Similar shapes

1. Similarity

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Are the polygons here similar to one another (select from the choices below)?
  2. Select the correct reason which explains the first answer.
polygon A polygon B
Answer:
  1. Are the shapes similar?
  2. The reason is:
HINT: <no title>
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Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two polygons
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The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

These polygons are similar: the angles are all 90 degrees, and the sides are proportional because the sides of each shape are all equal: 927=927. In fact, any two equilateral squares must be similar.

polygon A polygon B

These two polygons are similar. The correct choice from the list is: Yes.


STEP: Question 2: Choose the correct reason or reasons
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As noted in the above explanation, the shapes are similar. This can only be true if the corresponding angles are equal and the sides are all proportional: both must be true for similarity.

Therefore, the correct choice from the list is: Both of the above


Submit your answer as: and

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Gift has already answered the question, and his proof is written below. But, Gift has made a mistake. Look carefully at his proof and identify where he has made his mistake.

Line Proof
(a) In ΔDFE and ΔFGE:
(b) 1. EGEF=1510=1,5
(c) 2. DEFE=22,515=1,5
(d) 3. DFFG=1812=1,5
(e) ΔDFE|||ΔFGE (sides of Δ in prop)
Answer:

The mistake is on Line .

Replace this line with

HINT: <no title>
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Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
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When we use sides of Δ in prop to prove that two triangles are similar, we must divide the sides in the same way each time.

Here, Gift divided the sides in ΔDFE by the sides in ΔFGE on Lines (c) and (d). But, on Line (b), Gift wrote the sides the other way around. So, the mistake is on Line (b).

NOTE: Gift substituted in the values for EFEG, which is why he got the correct answer for the proportionality constant. But, this substitution did not match the labels he had written.

STEP: Correct the proof
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Gift should have divided the sides of ΔDFE by the sides of ΔFGE every time.

The correct proof is:

Line Proof
(a) In ΔDFE and ΔFGE:
(b) 1. EFEG=1510=1,5
(c) 2. DEFE=22,515=1,5
(d) 3. DFFG=1812=1,5
(e) ΔDFE|||ΔFGE (sides of Δ in prop)

Submit your answer as: and

Consequences of similarity

In the diagram below, ΔABC|||ΔRQS.

Determine the size of Q^.

Answer: Q^= °
numeric
HINT: <no title>
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If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
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If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so they must be equiangular. We use this fact to determine the size of Q^.

The angles in the second triangle must match the order of the angles in the first triangle, according to the similarity statement ΔABC|||ΔRQS. A and R are the first letters, so they are a matching pair. B and Q are the second letters, so they are a matching pair. C and S are the last letters, so they are a matching pair.

Since B^ and Q^ match up, they must be equal. This is only true because we were told that ΔABC|||ΔRQS. So, we use this as our reason.

Q^=77°(ΔABC|||ΔRQS)

Submit your answer as: and

Identify similar triangles

In the diagram below, EDDC and ECDA. Also, C^=x, DC=6, and AC=3.

  1. Ajibike needs to prove that ΔEDC|||ΔDAC. She has already started, and her incomplete proof is written below:

    In ΔEDC and ΔDAC:

    1. C^ is common
    2. ED^C=DA^C=90° (given)
      ...

    How should Ajibike complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
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    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
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    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Ajibike has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔEDC, we can write E^ in terms of x:

    E^+x+90°=180°(sum of s in Δ)E^+x+90°90°=180°90°E^+x=90°E^+xx=90°xE^=90°x

    In the same way in ΔDAC, we can write CD^A in terms of x.

    CD^A+x+90°=180°(sum of s in Δ)CD^A=90°x
    NOTE: We cannot assume that AD cuts ED^C in half. CD^A might be equal to 45°, or it might be something else. We must only use the information that we have been given.

    The full proof is shown below:

    In ΔEDC and ΔDAC:

    1. C^ is common
    2. ED^C=DA^C=90° (given)
    3. E^=90°x (sum of s in Δ)
      Also, CD^A=90°x (sum of s in Δ)
      E^=CD^A (both equal to 90°x)

    ΔEDC|||ΔDAC (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of EA.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: EA=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
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    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects EA to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
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    We are trying to find EA. Even though EA is not a side in the similar triangles, it is part of side EC:

    EC=EA+AC

    To make the algebra easier, let EA=y.

    Now we have

    EC=y+3

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that EC matches with DC: ΔEDC|||ΔDAC.

    We know the lengths of DC and AC. They are a matching pair of sides: (ΔEDC|||ΔDAC).

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    ECDC=DCAC(ΔEDC|||ΔDAC)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
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    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+36=63y+36×6=63×6y+3=363y+33=3633y=9

    Submit your answer as: and

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
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What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
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We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔABC and ΔDEF below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔABC have been multiplied by 2 to get ΔDEF.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔABC and ΔDEF.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

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Using the proportionality of similar polygons

In the diagram below, PQTU|||RSUW. What is the value of g? The diagram is not necessarily drawn to scale.

Answer: g=
numeric
HINT: <no title>
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Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for g.


STEP: Identify corresponding sides and write a proportion for these sides
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We have a diagram with quadrilaterals, and we need to figure out the value of the variable g. The question tells us that the quadrilaterals are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for g.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

PQRW=TUSUPQTU=RWSU

STEP: Solve the equation for g
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We will use the first proportion from above to find the value of g. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for g (and full marks).

Remember:

SU=TU+ST=2g+2+12=2g+14

So now we can calculate g:

PQRW=TUSUgg+3=2g+22g+14(g)(2g+14)=(2g+2)(g+3)2g2+14g=2g24g+6cancel2g2 terms18g=6g=618g=13

Therefore, the value of the variable is g=13.


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Given similar triangles, calculate a side

In the diagram below, ΔEAB|||ΔDAC. Also, BE=8, CD=3, CB=7, and AC=y .

Determine the value of y, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: y= units
one-of
type(numeric.abserror(0.005))
HINT: <no title>
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Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects y to the sides that you already know. You may have y more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
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If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find y. We can see that y is equal to AC. We can also see that:

AB=y+7

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that AB matches with AC: ΔEAB|||ΔDAC.

We know the lengths of EB and DC. They are a matching pair of sides: ΔEAB|||ΔDAC.

So, dividing the sides in the big triangle by those in the small triangle, we get

ABAC=EBDC(ΔEAB|||ΔDAC)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
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First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

y+7y=83y+7y×y=83×yy+7=8y3(y+7)×3=8y3×33y+21=8y3y+213y=8y3y21=5yy=215

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Given similar triangles, calculate a side

  1. In the diagram below, ΔBAD|||ΔCAB. Also, BA=4 and CA=3.

    Chibueze has been asked to determine the length of AD. He knows that in order to calculate AD, he first needs to match up the sides of the two similar triangles.

    Help Chibueze to match up the sides in the similar triangles.

    Answer:
    Side in ΔBAD Matching side in ΔCAB
    BA
    BD
    AD
    HINT: <no title>
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    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
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    Since ΔBAD|||ΔCAB, we can see that BA matches with CA. These are the first two letters in the similarity statement.

    Also using ΔBAD|||ΔCAB, we can see that BD matches with CB. These are the first and last letters in the similarity statement.

    Finally, we can use ΔBAD|||ΔCAB to work out that AD matches with AB. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of AD.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: AD= units
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
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    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
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    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for AD. So, start by writing AD in the numerator:

    AD=

    Since ΔBAD|||ΔCAB, we can see that AD matches with AB.

    So we have

    ADAB=

    We were also given information about BA and CA. Since ΔBAD|||ΔCAB, we can see that BA matches with CA.

    So,

    ADAB=BACA(ΔBAD|||ΔCAB)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
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    AD4=43AD=5,33333... units5,33 units

    Submit your answer as: and

Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔPQR and ΔXYZ?

HINT: <no title>
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The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
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Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

We can see that the sides in ΔPQR have been multiplied by 0,5 to get the matching sides in ΔXYZ.

So, the triangles are similar, and we write:

ΔPQR|||ΔXYZ (sides of Δ in prop).


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Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔCAB been multiplied by the same number, k, to get the sides in ΔEFD. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
HINT: <no title>
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Divide the longest side in ΔEFD by the longest side in ΔCAB.


STEP: Divide the longest side in ΔEFD by the longest side in ΔCAB
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We are looking for the number that each matching side in ΔCAB has been multiplied by, to get the matching side in ΔEFD.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔCAB is 14. It must match up with the longest side in ΔEFD, which is 21. We will use these sides to determine the multiplier. Let the proportionality constant be k:

14×k=2114×k14=2114k=1,5
NOTE: In the last step, we divided the longest side in ΔEFD by the longest side in ΔCAB, to find k.

We can check that this value works for the other sides:

  • The shortest sides, CA and EF: 6×1,5=9
  • The middle sides, AB and FD: 10×1,5=15

So, all the sides in ΔCAB have been multiplied by 1,5 to get the sides in ΔEFD. We call this number the proportionality constant.


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Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
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  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
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Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

The matching sides and angles in the two polygons are exactly equal.

So, the two polygons are congruent.

NOTE: Congruent polygons are always also similar to each other. But if two polygons are exactly equal, it is more correct to call them congruent.

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Identify similar triangles

ΔBAC is drawn below.

Which of the following triangles is definitely similar to ΔBAC? Give a reason for the similarity.

Answer:

ΔBAC|||Δ

string
HINT: <no title>
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Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
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Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

If we divide the matching sides in ΔRPQ by the matching sides in ΔBAC, we get the same number each time:

longest sides: RQBC=168=2middle sides: PQAC=147=2shortest sides: RPBA=126=2

So, these triangles are similar, because their sides are in proportion.


STEP: Match up the triangle vertices
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We must name the triangles in the order that their sides match up.

In ΔBAC, Point B is between the longest and the shortest sides. In the same way, in ΔRPQ Point R is between the longest and the shortest sides.

So, Point B matches with Point R.

In ΔBAC, Point A is the other end of the shortest side. In the same way, in ΔRPQ Point P is the other end of the shortest side. So, Point A matches with Point P.

Then, the last vertices must also match up. So, Point C matches with Point Q, and we write:

ΔBAC|||ΔRPQ (sides of Δ in prop).


Submit your answer as: and

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔABC|||ΔEFD.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔABC and ΔEFD:

  1. EF÷ =
  2. FD÷ =
  3. ED÷ =

ΔABC|||ΔEFD (sides of Δ in prop)

numeric
numeric
numeric
HINT: <no title>
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There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
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There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (EF and AB) match up.
  • The shortest sides (ED and AC) match up.
  • DF and BC must match up because they are the only sides left.

Now we will work out what each side in ΔABC should be multiplied by:

EFAB=6328=2,25FDBC=5424=2,25EDAC=2712=2,25

This means that any side in ΔABC must be multiplied by 2,25 to make its matching side in ΔEFD.

For example, 28×2,25=63.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
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Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔABC and ΔEFD:

  1. EF÷AB = 6328 = 2,25
  2. FD÷BC = 5424 = 2,25
  3. ED÷AC = 2712 = 2,25

ΔABC|||ΔEFD (sides of Δ in prop)


Submit your answer as: andandandandand

Basic deductions from similar triangles

  1. In the diagram below, ΔRPQ|||ΔEFD, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • Q= (ΔRPQ|||ΔEFD)
    • P= (ΔRPQ|||ΔEFD)
    • R= (ΔRPQ|||ΔEFD)
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
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    We have been told that ΔRPQ is similar to ΔEFD, because the sides are in proportion. We can see that the all of the sides in ΔRPQ have been multiplied by 3 to get the sides in ΔEFD.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔRPQ|||ΔEFD.

    So,

    • R^=E^(ΔRPQ|||ΔEFD)
    • P^=F^(ΔRPQ|||ΔEFD)
    • Q^=D^(ΔRPQ|||ΔEFD)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔNLM|||ΔCAB, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
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    We have been told that ΔNLM is similar to ΔCAB, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔNLM|||ΔCAB, we can see that NL matches with CA.

    In the same way, we can see that NM matches with CB (ΔNLM|||ΔCAB).

    Finally, the last sides must also match up. So LM matches with AB (ΔNLM|||ΔCAB).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔCAB's sides in the numerator for the first fraction, then ΔCAB's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    CBNM=CANL=ABLM(ΔNLM|||ΔCAB)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. NMCB=NLCA=LMAB(ΔNLM|||ΔCAB)
    2. CANL=ABLM=CBNM(ΔNLM|||ΔCAB)

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Identify similar triangles

In the diagram below, RSSP and SQRP. Also, R^=51° and P^=39°.

Identify one triangle that is similar to ΔRSP.

Answer: ΔRSP|||Δ
one-of
type(string.nocase)
HINT: <no title>
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Which other triangle in the diagram has the same angles as ΔRSP?


STEP: Identify the similar triangle
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There are three triangles in the diagram: ΔRSP, ΔSQP, and ΔRQS. They are all similar to each other.

Looking at ΔRSP and ΔSQP, we can see that:

  • P^ is a common angle.
  • RS^P=PQ^S=90°, which was given.
  • PS^Q=R^=51°, using sum of angles in ΔSQP.

So, ΔRSP|||ΔSQP, because the angles in both triangles are equal.

We could also prove that ΔRSP|||ΔRQS.

Since ΔSQP and ΔRQS are both similar to ΔRSP, they are also similar to each other.

So, either ΔRQS or ΔSQP is a correct answer.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

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Prove similarity with parallel lines

In the diagram below, TPSQ.

Prove that ΔTPR|||ΔSQR.

Answer:

In ΔTPR and ΔSQR:

  1. RPT= (corresp s; TPSQ)
  2. RTP= (corresp s; TPSQ)

ΔTPR|||Δ .

string
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
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There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔTPR and ΔSQR:

  1. RPT=RQS (corresp s; TPSQ)
  2. RTP=RSQ (corresp s; TPSQ)
  3. R is common

STEP: Complete the proof by labelling the triangles and giving a reason
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We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at T is equal to the angle at S. So, Point T matches with Point S.

In the same way:

  • Point P matches with Q, and
  • Point R matches with R.

So ΔTPR|||ΔSQR (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
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There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the sides of all the triangles and so we need to find a diagram that shows a pair of triangles with all three pairs of corresponding sides in proportion.

We look at the side labels. In Diagram A we note that the three pairs of corresponding sides are in different proportions. In Diagram B we note the three pairs of corresponding sides are in proportion.

Therefore Diagram B gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram B, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is B.


Submit your answer as:

Prove simple similarity

Consider the triangles given below:

Select the correct options to prove that ΔKLJ|||ΔDFE.

Answer:

In ΔKLJ and ΔDFE:

Step Reason
1. L=F=55° (given)
2. J=65° (given)
=65°
3. D=60° (given)
=60°
ΔKLJ|||ΔDFE
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

In ΔLJK:

K^=180°55°65° (sum of s in Δ)
K^=60°

In ΔFED:

E^=180°55°60° (sum of s in Δ)
E^=65°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔKLJ and ΔDFE:

  1. L^=F^=55° (given)
  2. J^=65° (given)
    E^=65° (sum of s in Δ)
    J^=E^
  3. D^=60° (given)
    K^=60° (sum of s in Δ)
    D^=K^

ΔKLJ|||ΔDFE (equiangular Δs)


Submit your answer as: andandandandandand

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is similar to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other.


STEP: Choose the similar polygon
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Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other. For two polygons to be similar, they must have:

  • the same number of sides,
  • all of the matching sides in proportion, and
  • all of the matching angles equal in size.
NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Option D is the same shape as the given polygon, even though it is a different size.

So, Option D is similar to the given polygon.


Submit your answer as:

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of n.

Answer: n= units
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
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If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 12 to give the matching sides in the second polygon. For example:

  • 10×12=5
  • 8×12=4

The side labelled n matches with the side equal to 16 in the first polygon. So:

16×12=nn=8 units
NOTE: We can see that the sides in the first polygon have all been divided by 2 to get the sides in the second polygon. Dividing by 2 is the same as multiplying by 12.

Submit your answer as:

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Habubakar needs to prove that ΔEFD|||ΔXYZ. He has already started, and his incomplete proof is written below:

    In ΔEFD and ΔXYZ:

    1. E^=X^ (given)
      ...
    Answer:

    How should Habubakar complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option B

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option C

    We cannot prove that the triangles are congruent, even though we have been given a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. Even though the triangles look almost the same size, the information we have been given actually proves that they are slightly different sizes.

    Option D

    This "proof" makes a mistake on Step 3. D^ is equal to Z^ because of sum of s in Δ. It was not given.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔEFD and ΔXYZ:

    1. E^=X^ (given)
    2. F^=Y^ (given)
    3. D^=81° (sum of s in Δ)
      Also Z^=81° (sum of s in Δ)
      D^=Z^

    ΔEFD|||ΔXYZ (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of XY, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: XY=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for XY. So, start by writing XY in the numerator:

    XY=

    Since ΔEFD|||ΔXYZ, we can see that XY matches with EF.

    So we have:

    XYEF=

    We were also given information about ZX and ED. Since ΔEFD|||ΔXYZ, we can see that ZX matches with ED.

    So,

    XYEF=ZXED(ΔEFD|||ΔXYZ)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    XY15=1512XY15×15=1512×15XY=754=18,75

    Submit your answer as: and

Exercises

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Determine whether the two shapes shown are similar or not.
  2. Select the correct reason which explains the first answer.
shape A shape B
Answer:
  1. Are the shapes similar?
  2. The reason is:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two shapes
[−1 point ⇒ 1 / 2 points left]

The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

These shape are parallelograms, and they are similar. We can see that the four angles are all equal, and the sides are all proportional: 46=1015.

shape A shape B

These two shapes are similar. The correct choice from the list is: Yes.


STEP: Question 2: Choose the correct reason or reasons
[−1 point ⇒ 0 / 2 points left]

As noted in the above explanation, the shapes are similar. This can only be true if the corresponding angles are equal and the sides are all proportional: both must be true for similarity.

Therefore, the correct choice from the list is: Both of the above


Submit your answer as: and

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Are the shapes here similar to one another (select from the choices below)?
  2. Select the correct reason which explains the first answer.
shape A shape B
Answer:
  1. Are the shapes similar?
  2. The reason is:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two shapes
[−1 point ⇒ 1 / 2 points left]

The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

These shapes are similar: the angles are all 90 degrees, and the sides are proportional because the sides of each shape are all equal: 36=36. In fact, any two equilateral squares must be similar.

shape A shape B

These two shapes are similar. The correct choice from the list is: Yes.


STEP: Question 2: Choose the correct reason or reasons
[−1 point ⇒ 0 / 2 points left]

As noted in the above explanation, the shapes are similar. This can only be true if the corresponding angles are equal and the sides are all proportional: both must be true for similarity.

Therefore, the correct choice from the list is: Both of the above


Submit your answer as: and

Identifying similar polygons

The diagram below shows two polygons. The figure is drawn to scale.

  1. Are the polygons here similar to one another (select from the choices below)?
  2. What is the ratio of the size of polygon A to polygon B. Give your answer as a fraction. If the polygons are not similar, write 'None' for the ratio.
polygon A polygon B
Answer:
  1. Are the shapes similar?
  2. The ratio of the lengths is:
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Similar objects must have the same shape, but might not be the same size.


STEP: Question 1: Compare the angles and sides of the two polygons
[−1 point ⇒ 1 / 2 points left]

The question shows us two shapes, and we need to decide if they are similar or not. In maths, two shapes are similar if corresponding angles are equal and if corresponding sides are all proportional (have the same ratio). In other words, if two objects are similar to each other, one of them can be "zoomed in or out" to make it identical to the other one.

These polygons are similar. In fact, they are equal (identical). If two objects are equal then they must be similar because the angles and sides are all equal.

polygon A polygon B

These two polygons are similar. The correct choice from the list is: Yes.


STEP: Question 2: Calculate the ratio of the shapes (if they are similar)
[−1 point ⇒ 0 / 2 points left]

Similar shapes always have proportional sides. In this case, the polygons have exactly the same shape, so the ratio of the sizes is 1.

The correct answer is: 1.


Submit your answer as: and

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Chukuemeka has already answered the question, and his proof is written below. But, Chukuemeka has made a mistake. Look carefully at his proof and identify where he has made his mistake.

Line Proof
(a) In ΔLJM and ΔLKJ:
(b) 1. MJJK=2012=53
(c) 2. MLJL=2515=53
(d) 3. LKLJ=159=53
(e) ΔLJM|||ΔLKJ (sides of Δ in prop)
Answer:

The mistake is on Line .

Replace this line with

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
[−1 point ⇒ 2 / 3 points left]

When we use sides of Δ in prop to prove that two triangles are similar, we must divide the sides in the same way each time.

Here, Chukuemeka divided the sides in ΔLJM by the sides in ΔLKJ on Lines (b) and (c). But, on Line (d), Chukuemeka wrote the sides the other way around. So, the mistake is on Line (d).

NOTE: Chukuemeka substituted in the values for LJLK, which is why he got the correct answer for the proportionality constant. But, this substitution did not match the labels he had written.

STEP: Correct the proof
[−2 points ⇒ 0 / 3 points left]

Chukuemeka should have divided the sides of ΔLJM by the sides of ΔLKJ every time.

The correct proof is:

Line Proof
(a) In ΔLJM and ΔLKJ:
(b) 1. MJJK=2012=53
(c) 2. MLJL=2515=53
(d) 3. LJLK=159=53
(e) ΔLJM|||ΔLKJ (sides of Δ in prop)

Submit your answer as: and

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Khayone has already answered the question, and his proof is written below. But, Khayone has made a mistake. Look carefully at his proof and identify where he has made his mistake.

Line Proof
(a) In ΔFEG and ΔEDG:
(b) 1. EGFG=2515=53
(c) 2. FEED=2012=53
(d) 3. GEGD=159=53
(e) ΔFEG|||ΔEDG (sides of Δ in prop)
Answer:

The mistake is on Line .

Replace this line with

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
[−1 point ⇒ 2 / 3 points left]

When we use sides of Δ in prop to prove that two triangles are similar, we must divide the sides in the same way each time.

Here, Khayone divided the sides in ΔFEG by the sides in ΔEDG on Lines (c) and (d). But, on Line (b), Khayone wrote the sides the other way around. So, the mistake is on Line (b).

NOTE: Khayone substituted in the values for FGEG, which is why he got the correct answer for the proportionality constant. But, this substitution did not match the labels he had written.

STEP: Correct the proof
[−2 points ⇒ 0 / 3 points left]

Khayone should have divided the sides of ΔFEG by the sides of ΔEDG every time.

The correct proof is:

Line Proof
(a) In ΔFEG and ΔEDG:
(b) 1. FGEG=2515=53
(c) 2. FEED=2012=53
(d) 3. GEGD=159=53
(e) ΔFEG|||ΔEDG (sides of Δ in prop)

Submit your answer as: and

Prove similarity in adjacent triangles

Consider the following diagram:

Prove that the triangles are similar.

Khadija has already answered the question, and her proof is written below. But, Khadija has made a mistake. Look carefully at her proof and identify where she has made her mistake.

Line Proof
(a) In ΔLMJ and ΔJMK:
(b) 1. MJMK=129=43
(c) 2. LMJM=1612=43
(d) 3. LJJK=2015=43
(e) ΔLMJ|||ΔJMK (similar Δs)
Answer:

The mistake is on Line .

Replace this line with

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Check each line of the proof carefully. Are the sides of the triangles being divided in a consistent order? Is the similarity stated correctly, with the correct reason?


STEP: Identify the error in the proof
[−1 point ⇒ 2 / 3 points left]

Khadija has proved that the sides of these two triangles are in proportion. But, on Line (e), Khadija has said that the two triangles were similar because of similar Δs. This is not the geometry reason Khadija used to prove similarity. So, the mistake is on Line (e).

TIP: In geometry, we have to use exact reasons.

STEP: Correct the proof
[−2 points ⇒ 0 / 3 points left]

Khadija should have used the correct reason: sides of Δ in prop.

The correct proof is:

Line Proof
(a) In ΔLMJ and ΔJMK:
(b) 1. MJMK=129=43
(c) 2. LMJM=1612=43
(d) 3. LJJK=2015=43
(e) ΔLMJ|||ΔJMK (sides of Δ in prop)

Submit your answer as: and

Consequences of similarity

In the diagram below, ΔDEF|||ΔJLK.

Determine the size of L^.

Answer: L^= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
[−2 points ⇒ 0 / 2 points left]

If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so they must be equiangular. We use this fact to determine the size of L^.

The angles in the second triangle must match the order of the angles in the first triangle, according to the similarity statement ΔDEF|||ΔJLK. D and J are the first letters, so they are a matching pair. E and L are the second letters, so they are a matching pair. F and K are the last letters, so they are a matching pair.

Since E^ and L^ match up, they must be equal. This is only true because we were told that ΔDEF|||ΔJLK. So, we use this as our reason.

L^=48°(ΔDEF|||ΔJLK)

Submit your answer as: and

Consequences of similarity

In the diagram below, ΔQRS|||ΔZYX.

Determine the length of ZX.

Answer: ZX= units
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
[−2 points ⇒ 0 / 2 points left]

If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so their sides will be in proportion. We use this fact to determine the length of ZX.

First we match up the sides. In the statement ΔQRS|||ΔZYX, QR and ZY are the first two letters, so their sides must match up. Similarly, RS and YX match up because their letters came last. Finally, QS and ZX must match up, because they are the only sides left.

Since QR and ZY match up, we can work out the proportionality constant:

k=ZYQRk=5424k=2,25

This is the number by which all sides in ΔQRS should by multiplied in order to get the matching sides in ΔZYX, because 24×2,25=54.

This is only true because we were told that ΔQRS|||ΔZYX. So, we use this as our reason.

So,

ZX=16×2,25ZX=36 units(ΔQRS|||ΔZYX)

Submit your answer as: and

Consequences of similarity

In the diagram below, ΔMNP|||ΔZXY.

Determine the length of ZY.

Answer: ZY= units
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two triangles are similar, then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

STEP: Match up the sides and angles
[−2 points ⇒ 0 / 2 points left]

If two triangles are similar then

  1. they are equiangular (their matching angles are equal), and
  2. their sides are in proportion (you always have to multiply by the same number to get the matching side in the other triangle).

We know that the triangles are similar, so their sides will be in proportion. We use this fact to determine the length of ZY.

First we match up the sides. In the statement ΔMNP|||ΔZXY, MN and ZX are the first two letters, so their sides must match up. Similarly, NP and XY match up because their letters came last. Finally, MP and ZY must match up, because they are the only sides left.

Since MN and ZX match up, we can work out the proportionality constant:

k=ZXMNk=7040k=1,75

This is the number by which all sides in ΔMNP should by multiplied in order to get the matching sides in ΔZXY, because 40×1,75=70.

This is only true because we were told that ΔMNP|||ΔZXY. So, we use this as our reason.

So,

ZY=36×1,75ZY=63 units(ΔMNP|||ΔZXY)

Submit your answer as: and

Identify similar triangles

In the diagram below, TSSP and TPSR. Also, P^=x, SP=8, and RP=5.

  1. Jabulile needs to prove that ΔTSP|||ΔSRP. She has already started, and her incomplete proof is written below:

    In ΔTSP and ΔSRP:

    1. P^ is common
    2. TS^P=SR^P=90° (given)
      ...

    How should Jabulile complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
    [−3 points ⇒ 0 / 3 points left]
    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Jabulile has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔTSP, we can write T^ in terms of x:

    T^+x+90°=180°(sum of s in Δ)T^+x+90°90°=180°90°T^+x=90°T^+xx=90°xT^=90°x

    In the same way in ΔSRP, we can write PS^R in terms of x.

    PS^R+x+90°=180°(sum of s in Δ)PS^R=90°x
    NOTE: We cannot assume that RS cuts TS^P in half. PS^R might be equal to 45°, or it might be something else. We must only use the information that we have been given.

    The full proof is shown below:

    In ΔTSP and ΔSRP:

    1. P^ is common
    2. TS^P=SR^P=90° (given)
    3. T^=90°x (sum of s in Δ)
      Also, PS^R=90°x (sum of s in Δ)
      T^=PS^R (both equal to 90°x)

    ΔTSP|||ΔSRP (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of TR.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: TR=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects TR to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 2 / 4 points left]

    We are trying to find TR. Even though TR is not a side in the similar triangles, it is part of side TP:

    TP=TR+RP

    To make the algebra easier, let TR=y.

    Now we have

    TP=y+5

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that TP matches with SP: ΔTSP|||ΔSRP.

    We know the lengths of SP and RP. They are a matching pair of sides: (ΔTSP|||ΔSRP).

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    TPSP=SPRP(ΔTSP|||ΔSRP)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
    [−2 points ⇒ 0 / 4 points left]

    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+58=85y+58×8=85×8y+5=645y+55=6455y=395

    Submit your answer as: and

Identify similar triangles

In the diagram below, CEEA and CAED. Also, A^=x, EA=6, and DA=5.

  1. Khayalethu needs to prove that ΔCEA|||ΔEDA. He has already started, and his incomplete proof is written below:

    In ΔCEA and ΔEDA:

    1. A^ is common
    2. CE^A=ED^A=90° (given)
      ...

    How should Khayalethu complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
    [−3 points ⇒ 0 / 3 points left]
    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Khayalethu has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔCEA, we can write C^ in terms of x:

    C^+x+90°=180°(sum of s in Δ)C^+x+90°90°=180°90°C^+x=90°C^+xx=90°xC^=90°x

    In the same way in ΔEDA, we can write AE^D in terms of x.

    AE^D+x+90°=180°(sum of s in Δ)AE^D=90°x
    NOTE: We cannot assume that DE cuts CE^A in half. AE^D might be equal to 45°, or it might be something else. We must only use the information that we have been given.

    The full proof is shown below:

    In ΔCEA and ΔEDA:

    1. A^ is common
    2. CE^A=ED^A=90° (given)
    3. C^=90°x (sum of s in Δ)
      Also, AE^D=90°x (sum of s in Δ)
      C^=AE^D (both equal to 90°x)

    ΔCEA|||ΔEDA (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of CD.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: CD=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects CD to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 2 / 4 points left]

    We are trying to find CD. Even though CD is not a side in the similar triangles, it is part of side CA:

    CA=CD+DA

    To make the algebra easier, let CD=y.

    Now we have

    CA=y+5

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that CA matches with EA: ΔCEA|||ΔEDA.

    We know the lengths of EA and DA. They are a matching pair of sides: (ΔCEA|||ΔEDA).

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    CAEA=EADA(ΔCEA|||ΔEDA)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
    [−2 points ⇒ 0 / 4 points left]

    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+56=65y+56×6=65×6y+5=365y+55=3655y=115

    Submit your answer as: and

Identify similar triangles

In the diagram below, TQQP and TPQR. Also, P^=x, QP=8, and RP=5.

  1. Daniel needs to prove that ΔTQP|||ΔQRP. He has already started, and his incomplete proof is written below:

    In ΔTQP and ΔQRP:

    1. P^ is common
    2. TQ^P=QR^P=90° (given)
      ...

    How should Daniel complete the proof? Choose the best option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Which can you use here?


    STEP: Choose the correct proof
    [−3 points ⇒ 0 / 3 points left]
    NOTE:

    There are two ways of proving similarity in triangles:

    • Prove that the two triangles are equiangular.
    • Prove that the two triangles have matching sides in proportion.

    Daniel has already proved two pairs of angles equal for us, and we need to prove the third pair. To do so, it is helpful to look at each triangle separately.

    Looking at ΔTQP, we can write T^ in terms of x:

    T^+x+90°=180°(sum of s in Δ)T^+x+90°90°=180°90°T^+x=90°T^+xx=90°xT^=90°x

    In the same way in ΔQRP, we can write PQ^R in terms of x.

    PQ^R+x+90°=180°(sum of s in Δ)PQ^R=90°x
    NOTE: We cannot assume that RQ cuts TQ^P in half. PQ^R might be equal to 45°, or it might be something else. We must only use the information that we have been given.

    The full proof is shown below:

    In ΔTQP and ΔQRP:

    1. P^ is common
    2. TQ^P=QR^P=90° (given)
    3. T^=90°x (sum of s in Δ)
      Also, PQ^R=90°x (sum of s in Δ)
      T^=PQ^R (both equal to 90°x)

    ΔTQP|||ΔQRP (equiangular Δs)


    Submit your answer as:
  2. Hence, determine the length of TR.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Give your answer as a fraction, where necessary.
    Answer: TR=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects TR to the sides that you already know. Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 2 / 4 points left]

    We are trying to find TR. Even though TR is not a side in the similar triangles, it is part of side TP:

    TP=TR+RP

    To make the algebra easier, let TR=y.

    Now we have

    TP=y+5

    The similarity is easier to see if we separate the triangles.

    We can see from the similarity statement that TP matches with QP: ΔTQP|||ΔQRP.

    We know the lengths of QP and RP. They are a matching pair of sides: (ΔTQP|||ΔQRP).

    So, dividing the sides of the big triangle by the sides of the small triangle, we get:

    TPQP=QPRP(ΔTQP|||ΔQRP)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Solve the equation using inverse operations
    [−2 points ⇒ 0 / 4 points left]

    First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

    y+58=85y+58×8=85×8y+5=645y+55=6455y=395

    Submit your answer as: and

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
[−2 points ⇒ 0 / 2 points left]

We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔDEF and ΔJKL below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔDEF have been multiplied by 0,5 to get ΔJKL.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔDEF and ΔJKL.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

Submit your answer as:

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
[−2 points ⇒ 0 / 2 points left]

We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔABC and ΔDEF below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔABC have been multiplied by 0,5 to get ΔDEF.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔABC and ΔDEF.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

Submit your answer as:

Definition: Similarity

Two triangles are similar if:

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

What do we mean in maths when we say that two triangles are similar?


STEP: Identify the definition of similarity
[−2 points ⇒ 0 / 2 points left]

We say that two triangles are similar if they are exactly the same shape, but one might be bigger or smaller than the other one. Another way of saying this is that one is an enlargement of the other one.

For example, ΔPQR and ΔXYZ below are the same shape. But, they are different sizes.

The sides of similar triangles are in proportion. That means that all of the sides in one triangle have been multiplied by the same number to get the sides in the second triangle. All of the sides in ΔPQR have been multiplied by 0,5 to get ΔXYZ.

Similar triangles are the same shape, so they always have the same angles. We can see the matching angles in ΔPQR and ΔXYZ.

NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Two triangles are similar if either

  • their matching sides are in proportion, or
  • their matching angles are equal.
NOTE: In many polygons, we have to prove that the matching sides are in proportion and the matching angles are equal, if we want to be sure that the shapes are similar. But, in triangles, we only need to prove that one of them is true. This is a special property of triangles, which you will learn more about in grade 12.

Submit your answer as:

Using the proportionality of similar polygons

In the diagram below, RSXY|||TWYZ. Compute the value of f. The diagram is not necessarily drawn to scale.

Answer: f=
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for f.


STEP: Identify corresponding sides and write a proportion for these sides
[−1 point ⇒ 2 / 3 points left]

We have a diagram with quadrilaterals, and we need to figure out the value of the variable f. The question tells us that the quadrilaterals are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for f.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

RSTZ=XYWYRSXY=TZWY

STEP: Solve the equation for f
[−2 points ⇒ 0 / 3 points left]

We will use the first proportion from above to find the value of f. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for f (and full marks).

Remember:

WY=XY+WX=2f+3+4=2f+7

So now we can calculate f:

RSTZ=XYWY2f2f+12=2f+32f+7(2f)(2f+7)=(2f+3)(2f+12)4f2+14f=4f218f+36cancel4f2 terms32f=36f=3632f=98

Therefore, the value of the variable is f=98.


Submit your answer as:

Using the proportionality of similar polygons

Triangles PQT and PRS are similar. Compute the value of g. The diagram is not necessarily drawn to scale.

Answer: g=
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for g.


STEP: Identify corresponding sides and write a proportion for these sides
[−1 point ⇒ 2 / 3 points left]

We have a diagram with triangles, and we need to figure out the value of the variable g. The question tells us that the triangles are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for g.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

PQQR=PTSTPQPR=PTPS

STEP: Solve the equation for g
[−2 points ⇒ 0 / 3 points left]

We will use the first proportion from above to find the value of g. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for g (and full marks).

PQQR=PTST2g15=3g+45(2g)(5)=(3g+4)(15)10g=45g+6055g=60g=6055g=1211

Therefore, the value of the variable is g=1211.


Submit your answer as:

Using the proportionality of similar polygons

Triangles PQT and PRS are similar. What is the value of f? The diagram is not necessarily drawn to scale.

Answer: f=
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Similar shapes have proportional lengths. Write a proportion for the sides of the polygons and then solve the equation for f.


STEP: Identify corresponding sides and write a proportion for these sides
[−1 point ⇒ 2 / 3 points left]

We have a diagram with triangles, and we need to figure out the value of the variable f. The question tells us that the triangles are similar, which means that the sides of the shapes are proportional. Therefore, we can write a proportion for the sides of the polygons and solve it for f.

There are a number of different proportions that we can write to summarise the lengths of the sides in the diagram: we can write a proportion for any pairs of corresponding sides. Here are two proportions which are accurate (but there are others):

PQQR=PTSTPQPR=PTPS

STEP: Solve the equation for f
[−2 points ⇒ 0 / 3 points left]

We will use the first proportion from above to find the value of f. If you use a different correct proportion and solve it without any mistakes, you will get the correct answer for f (and full marks).

PQQR=PTST3f3=f+26(3f)(6)=(f+2)(3)18f=3f+615f=6f=615f=25

Therefore, the value of the variable is f=25.


Submit your answer as:

Given similar triangles, calculate a side

In the diagram below, ΔBED|||ΔCEA. Also, BD=8, CA=3, CB=7, and EC=y .

Determine the value of y, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: y= units
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects y to the sides that you already know. You may have y more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
[−2 points ⇒ 2 / 4 points left]

If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find y. We can see that y is equal to EC. We can also see that:

EB=y+7

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that BE matches with CE: ΔBED|||ΔCEA.

We know the lengths of BD and CA. They are a matching pair of sides: ΔBED|||ΔCEA.

So, dividing the sides in the big triangle by those in the small triangle, we get

BECE=BDCA(ΔBED|||ΔCEA)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
[−2 points ⇒ 0 / 4 points left]

First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

y+7y=83y+7y×y=83×yy+7=8y3(y+7)×3=8y3×33y+21=8y3y+213y=8y3y21=5yy=215

Submit your answer as: and

Given similar triangles, calculate a side

In the diagram below, ΔPTS|||ΔPRQ. Also, TS=8, RQ=3, RT=7, and PR=b .

Determine the value of b, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: b= units
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects b to the sides that you already know. You may have b more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
[−2 points ⇒ 2 / 4 points left]

If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find b. We can see that b is equal to PR. We can also see that:

PT=b+7

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that PT matches with PR: ΔPTS|||ΔPRQ.

We know the lengths of TS and RQ. They are a matching pair of sides: ΔPTS|||ΔPRQ.

So, dividing the sides in the big triangle by those in the small triangle, we get

PTPR=TSRQ(ΔPTS|||ΔPRQ)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
[−2 points ⇒ 0 / 4 points left]

First we need to substitute in the values from the diagram. As long as there are only numbers and b-terms, we will be able to solve the equation.

b+7b=83b+7b×b=83×bb+7=8b3(b+7)×3=8b3×33b+21=8b3b+213b=8b3b21=5bb=215

Submit your answer as: and

Given similar triangles, calculate a side

In the diagram below, ΔKLJ|||ΔMLN. Also, JK=9, NM=4, NJ=7, and LN=y .

Determine the value of y, giving a reason for your answer.

INSTRUCTION: Give your answer as a fraction, where necessary.
Answer: y= units
one-of
type(numeric.abserror(0.005))
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Since the triangles are similar, you know that the sides are in proportion. Use this fact to create an equation which connects y to the sides that you already know. You may have y more than once in your equation!


STEP: Use the similarity statement to choose the correct pair of ratios
[−2 points ⇒ 2 / 4 points left]

If two triangles are similar, their sides are in proportion. So, when you divide the sides of one triangle with the matching sides in the other triangle, you always get the same number.

TIP: Use the similarity statement to choose the correct pair of ratios by making pairs of matching sides.

We are trying to find y. We can see that y is equal to LN. We can also see that:

LJ=y+7

The similarity is easier to see if we separate the triangles.

We can see from the similarity statement that LJ matches with LN: ΔKLJ|||ΔMLN.

We know the lengths of KJ and MN. They are a matching pair of sides: ΔKLJ|||ΔMLN.

So, dividing the sides in the big triangle by those in the small triangle, we get

LJLN=KJMN(ΔKLJ|||ΔMLN)
NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

STEP: Solve the equation using inverse operations
[−2 points ⇒ 0 / 4 points left]

First we need to substitute in the values from the diagram. As long as there are only numbers and y-terms, we will be able to solve the equation.

y+7y=94y+7y×y=94×yy+7=9y4(y+7)×4=9y4×44y+28=9y4y+284y=9y4y28=5yy=285

Submit your answer as: and

Given similar triangles, calculate a side

  1. In the diagram below, ΔCDA|||ΔCBD. Also, BD=12, CD=13, and CB=5.

    Dumile has been asked to determine the length of DA. He knows that in order to calculate DA, he first needs to match up the sides of the two similar triangles.

    Help Dumile to match up the sides in the similar triangles.

    Answer:
    Side in ΔCDA Matching side in ΔCBD
    CD
    CA
    DA
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
    [−2 points ⇒ 0 / 2 points left]

    Since ΔCDA|||ΔCBD, we can see that CD matches with CB. These are the first two letters in the similarity statement.

    Also using ΔCDA|||ΔCBD, we can see that CA matches with CD. These are the first and last letters in the similarity statement.

    Finally, we can use ΔCDA|||ΔCBD to work out that DA matches with BD. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of DA.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: DA= units
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for DA. So, start by writing DA in the numerator:

    DA=

    Since ΔCDA|||ΔCBD, we can see that DA matches with BD.

    So we have

    DABD=

    We were also given information about CD and CB. Since ΔCDA|||ΔCBD, we can see that CD matches with CB.

    So,

    DABD=CDCB(ΔCDA|||ΔCBD)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
    [−1 point ⇒ 0 / 3 points left]
    DA12=135DA=31,2 units

    Submit your answer as: and

Given similar triangles, calculate a side

  1. In the diagram below, ΔSPQ|||ΔSRP. Also, SP=10 and SR=6.

    Delphino has been asked to determine the length of SQ. He knows that in order to calculate SQ, he first needs to match up the sides of the two similar triangles.

    Help Delphino to match up the sides in the similar triangles.

    Answer:
    Side in ΔSPQ Matching side in ΔSRP
    SP
    SQ
    PQ
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
    [−2 points ⇒ 0 / 2 points left]

    Since ΔSPQ|||ΔSRP, we can see that SP matches with SR. These are the first two letters in the similarity statement.

    Also using ΔSPQ|||ΔSRP, we can see that SQ matches with SP. These are the first and last letters in the similarity statement.

    Finally, we can use ΔSPQ|||ΔSRP to work out that PQ matches with RP. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of SQ.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: SQ= units
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for SQ. So, start by writing SQ in the numerator:

    SQ=

    Since ΔSPQ|||ΔSRP, we can see that SQ matches with SP.

    So we have

    SQSP=

    We were also given information about SP and SR. Since ΔSPQ|||ΔSRP, we can see that SP matches with SR.

    So,

    SQSP=SPSR(ΔSPQ|||ΔSRP)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
    [−1 point ⇒ 0 / 3 points left]
    SQ10=106SQ=16,66666... units16,67 units

    Submit your answer as: and

Given similar triangles, calculate a side

  1. In the diagram below, ΔDAC|||ΔBAD. Also, DA=12 and BA=5.

    Adeboye has been asked to determine the length of AC. He knows that in order to calculate AC, he first needs to match up the sides of the two similar triangles.

    Help Adeboye to match up the sides in the similar triangles.

    Answer:
    Side in ΔDAC Matching side in ΔBAD
    DA
    DC
    AC
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Use the similarity statement to choose the correct pairs of sides.


    STEP: Use the similarity statement to match the sides
    [−2 points ⇒ 0 / 2 points left]

    Since ΔDAC|||ΔBAD, we can see that DA matches with BA. These are the first two letters in the similarity statement.

    Also using ΔDAC|||ΔBAD, we can see that DC matches with BD. These are the first and last letters in the similarity statement.

    Finally, we can use ΔDAC|||ΔBAD to work out that AC matches with AD. These are the second and third letters in the similarity statement.

    Visually we can see the matching sides of the triangles:


    Submit your answer as: andand
  2. Determine the value of AC.

    The diagram is repeated here for your convenience.

    INSTRUCTION: Round your answer to two decimal places, if necessary.
    Answer: AC= units
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similaity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, then their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for AC. So, start by writing AC in the numerator:

    AC=

    Since ΔDAC|||ΔBAD, we can see that AC matches with AD.

    So we have

    ACAD=

    We were also given information about DA and BA. Since ΔDAC|||ΔBAD, we can see that DA matches with BA.

    So,

    ACAD=DABA(ΔDAC|||ΔBAD)
    NOTE: We only know that the fractions are equal because the triangles are similar. So, we write this as the reason.

    STEP: Substitute in the lengths of the sides and solve the equation
    [−1 point ⇒ 0 / 3 points left]
    AC12=125AC=28,8 units

    Submit your answer as: and

Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔDEF and ΔJKL?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
[−2 points ⇒ 0 / 2 points left]

Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

We can see that the sides in ΔDEF have been multiplied by 2 to get the matching sides in ΔJKL.

So, the triangles are similar, and we write:

ΔDEF|||ΔJKL (sides of Δ in prop).


Submit your answer as:

Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔDEF and ΔJKL?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
[−2 points ⇒ 0 / 2 points left]

Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

We were given one matching side and two matching angles in both triangles. So, the triangles are congruent, and we write:

ΔDEFΔJKL (SAA)

NOTE: If two triangles are congruent, then they are always also similar. But, being congruent is more special that being similar, so the best way to describe them is to say that they are congruent.

Submit your answer as:

Similarity and congruency notation

Consider the following triangles:

Answer:

Which of the following best describes the relationship between ΔDEF and ΔJKL?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The symbol is used to indicate that two triangles are congruent. The symbol ||| is used to indicate that two triangles are similar.


STEP: Identify the triangle relationship
[−2 points ⇒ 0 / 2 points left]

Two triangles are similar if they are the same shape, but they can be a different size. The symbol for similarity is |||. There are two ways to know if triangles are similar:

  • if their sides are in proportion. This means that the sides in one triangle have been multiplied by the same number to get the sides in another triangle.
  • if all of their matching angles are equal.

Two triangles are congruent if they are exactly the same size and shape. The symbol for congruency is . There are four ways to know if triangles are congruent: SSS, SAS, SAA, or 90°HS.

TIP: When two triangles are equal in all respects, we say that they are congruent and use the symbol for congruency (). We never use the equals sign (=) to compare two triangles.

We were given two sides and the included angle in both triangles. So, the triangles are congruent, and we write:

ΔDEFΔJKL (SAS)

NOTE: If two triangles are congruent, then they are always also similar. But, being congruent is more special that being similar, so the best way to describe them is to say that they are congruent.

Submit your answer as:

Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔKLJ been multiplied by the same number, k, to get the sides in ΔPMN. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Divide the longest side in ΔPMN by the longest side in ΔKLJ.


STEP: Divide the longest side in ΔPMN by the longest side in ΔKLJ
[−2 points ⇒ 0 / 2 points left]

We are looking for the number that each matching side in ΔKLJ has been multiplied by, to get the matching side in ΔPMN.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔKLJ is 24. It must match up with the longest side in ΔPMN, which is 30. We will use these sides to determine the multiplier. Let the proportionality constant be k:

24×k=3024×k24=3024k=1,25
NOTE: In the last step, we divided the longest side in ΔPMN by the longest side in ΔKLJ, to find k.

We can check that this value works for the other sides:

  • The shortest sides, KL and PM: 12×1,25=15
  • The middle sides, LJ and MN: 20×1,25=25

So, all the sides in ΔKLJ have been multiplied by 1,25 to get the sides in ΔPMN. We call this number the proportionality constant.


Submit your answer as:

Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔCBA been multiplied by the same number, k, to get the sides in ΔDFE. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Divide the longest side in ΔDFE by the longest side in ΔCBA.


STEP: Divide the longest side in ΔDFE by the longest side in ΔCBA
[−2 points ⇒ 0 / 2 points left]

We are looking for the number that each matching side in ΔCBA has been multiplied by, to get the matching side in ΔDFE.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔCBA is 14. It must match up with the longest side in ΔDFE, which is 21. We will use these sides to determine the multiplier. Let the proportionality constant be k:

14×k=2114×k14=2114k=1,5
NOTE: In the last step, we divided the longest side in ΔDFE by the longest side in ΔCBA, to find k.

We can check that this value works for the other sides:

  • The shortest sides, CB and DF: 6×1,5=9
  • The middle sides, BA and FE: 12×1,5=18

So, all the sides in ΔCBA have been multiplied by 1,5 to get the sides in ΔDFE. We call this number the proportionality constant.


Submit your answer as:

Determine the proportionality factor between similar triangles

The diagram below shows two similar triangles:

All of the sides in ΔKLJ been multiplied by the same number, k, to get the sides in ΔPMN. What is the value of k?

INSTRUCTION: Give your answer as an exact decimal or as a fraction.
Answer: k=
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Divide the longest side in ΔPMN by the longest side in ΔKLJ.


STEP: Divide the longest side in ΔPMN by the longest side in ΔKLJ
[−2 points ⇒ 0 / 2 points left]

We are looking for the number that each matching side in ΔKLJ has been multiplied by, to get the matching side in ΔPMN.

NOTE: We call this number the proportionality constant, k.

The longest side in ΔKLJ is 15. It must match up with the longest side in ΔPMN, which is 22,5. We will use these sides to determine the multiplier. Let the proportionality constant be k:

15×k=22,515×k15=22,515k=1,5
NOTE: In the last step, we divided the longest side in ΔPMN by the longest side in ΔKLJ, to find k.

We can check that this value works for the other sides:

  • The shortest sides, KL and PM: 9×1,5=13,5
  • The middle sides, LJ and MN: 14×1,5=21

So, all the sides in ΔKLJ have been multiplied by 1,5 to get the sides in ΔPMN. We call this number the proportionality constant.


Submit your answer as:

Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]
  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
[−1 point ⇒ 0 / 1 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

The matching sides and angles in the two polygons are exactly equal.

So, the two polygons are congruent.

NOTE: Congruent polygons are always also similar to each other. But if two polygons are exactly equal, it is more correct to call them congruent.

Submit your answer as:

Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]
  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
[−1 point ⇒ 0 / 1 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

The matching sides and angles in the two polygons are exactly equal.

So, the two polygons are congruent.

NOTE: Congruent polygons are always also similar to each other. But if two polygons are exactly equal, it is more correct to call them congruent.

Submit your answer as:

Similar, congruent, or neither?

Consider the following polygons:

Are the polygons similar, congruent, or neither? Choose the most correct description.

Answer:

The two polygons are:

HINT: <no title>
[−0 points ⇒ 1 / 1 points left]
  • Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal and all of their matching sides are in proportion.
  • Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal. If we cut out two congruent polygons, one could be placed exactly on top of the other.

STEP: Identify if the polygons are similar, congruent, or neither
[−1 point ⇒ 0 / 1 points left]
Two polygons are similar if they are the same shape, but they may be different sizes. All of their matching angles are equal in size and all of their matching sides are in proportion.
Two polygons are congruent if they are exactly the same size and shape. All their matching sides and angles are equal.

The two polygons do not have to be in the same orientation. This means that they can face different directions.

The matching angles in the two quadrilaterals are exactly the same. We can also see that all of the sides in the first quadrilateral have been multiplied by 0,5 to get the sides in the second one. This means that the sides are in proportion.

This is easier to see if the quadrilaterals are in the same orientation.

So, the two polygons are similar.


Submit your answer as:

Identify similar triangles

ΔCAB is drawn below.

Which of the following triangles is definitely similar to ΔCAB? Give a reason for the similarity.

Answer:

ΔCAB|||Δ

string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
[−1 point ⇒ 1 / 2 points left]

Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

If we divide the matching sides in ΔYXZ by the matching sides in ΔCAB, we get the same number each time:

longest sides: YZCB=147=2middle sides: XZAB=105=2shortest sides: YXCA=63=2

So, these triangles are similar, because their sides are in proportion.


STEP: Match up the triangle vertices
[−1 point ⇒ 0 / 2 points left]

We must name the triangles in the order that their sides match up.

In ΔCAB, Point C is between the longest and the shortest sides. In the same way, in ΔYXZ Point Y is between the longest and the shortest sides.

So, Point C matches with Point Y.

In ΔCAB, Point A is the other end of the shortest side. In the same way, in ΔYXZ Point X is the other end of the shortest side. So, Point A matches with Point X.

Then, the last vertices must also match up. So, Point B matches with Point Z, and we write:

ΔCAB|||ΔYXZ (sides of Δ in prop).


Submit your answer as: and

Identify similar triangles

ΔPQR is drawn below.

Which of the following triangles is definitely similar to ΔPQR? Give a reason for the similarity.

Answer:

ΔPQR|||Δ

string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
[−1 point ⇒ 1 / 2 points left]

Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

We were given three pairs of equal angles in the two triangles, so they are similar.


STEP: Match up the triangle vertices
[−1 point ⇒ 0 / 2 points left]

We must name the triangles in the order that their angles match up.

We can see that the angle at P is equal to the angle at Y. So, Point P matches with Point Y.

In the same way:

  • Point Q matches with Z, and
  • Point R matches with X.

So,

ΔPQR|||ΔYZX (equiangular Δs)

NOTE: Equiangular comes from the equal angles in both triangles. It looks similar to equilateral, which means equal lines. But, the two words do not mean the same thing.

Submit your answer as: and

Identify similar triangles

ΔRPQ is drawn below.

Which of the following triangles is definitely similar to ΔRPQ? Give a reason for the similarity.

Answer:

ΔRPQ|||Δ

string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Triangles are similar if either their:

  • matching sides are in proportion (sides of Δ in prop), or
  • matching angles are equal (equiangular Δs).

STEP: Identify the similar triangles
[−1 point ⇒ 1 / 2 points left]

Triangles are similar if either their:

  • sides are in proportion (sides of Δ in prop), or
  • angles are equal (equiangular Δs).

We were given three pairs of equal angles in the two triangles, so they are similar.


STEP: Match up the triangle vertices
[−1 point ⇒ 0 / 2 points left]

We must name the triangles in the order that their angles match up.

We can see that the angle at R is equal to the angle at A. So, Point R matches with Point A.

In the same way:

  • Point P matches with B, and
  • Point Q matches with C.

So,

ΔRPQ|||ΔABC (equiangular Δs)

NOTE: Equiangular comes from the equal angles in both triangles. It looks similar to equilateral, which means equal lines. But, the two words do not mean the same thing.

Submit your answer as: and

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔPQR|||ΔZYX.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔPQR and ΔZYX:

  1. ZY÷ =
  2. YX÷ =
  3. ZX÷ =

ΔPQR|||ΔZYX (sides of Δ in prop)

numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (ZY and PQ) match up.
  • The shortest sides (ZX and PR) match up.
  • XY and QR must match up because they are the only sides left.

Now we will work out what each side in ΔPQR should be multiplied by:

ZYPQ=2816=1,75YXQR=2112=1,75ZXPR=148=1,75

This means that any side in ΔPQR must be multiplied by 1,75 to make its matching side in ΔZYX.

For example, 16×1,75=28.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]

Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔPQR and ΔZYX:

  1. ZY÷PQ = 2816 = 1,75
  2. YX÷QR = 2112 = 1,75
  3. ZX÷PR = 148 = 1,75

ΔPQR|||ΔZYX (sides of Δ in prop)


Submit your answer as: andandandandand

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔBAC|||ΔEDF.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔBAC and ΔEDF:

  1. ED÷ =
  2. DF÷ =
  3. EF÷ =

ΔBAC|||ΔEDF (sides of Δ in prop)

numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (ED and BA) match up.
  • The shortest sides (EF and BC) match up.
  • FD and AC must match up because they are the only sides left.

Now we will work out what each side in ΔBAC should be multiplied by:

EDBA=5424=2,25DFAC=4520=2,25EFBC=2712=2,25

This means that any side in ΔBAC must be multiplied by 2,25 to make its matching side in ΔEDF.

For example, 24×2,25=54.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]

Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔBAC and ΔEDF:

  1. ED÷BA = 5424 = 2,25
  2. DF÷AC = 4520 = 2,25
  3. EF÷BC = 2712 = 2,25

ΔBAC|||ΔEDF (sides of Δ in prop)


Submit your answer as: andandandandand

Prove simple similarity with sides

Consider the following triangles:

Prove that ΔJKL|||ΔNMP.

INSTRUCTION: Give your answer as a simplified fraction or decimal.
Answer:

In ΔJKL and ΔNMP:

  1. NM÷ =
  2. MP÷ =
  3. NP÷ =

ΔJKL|||ΔNMP (sides of Δ in prop)

numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

Which of these strategies is more appropriate for this question?


STEP: Match up sides and find the proportionality constant
[−3 points ⇒ 1 / 4 points left]

There are two ways of proving that triangles are similar:

  1. Prove that they are equiangular.
  2. Prove that their sides are in proportion.

In this question, we have more information about the sides of the triangles than about the angles. So, we will prove that all of the sides are in proportion.

We must show that each side must be multiplied by the same number to make its matching side in the other triangle.

Firstly, we must match up the sides:

  • The longest sides (NM and JK) match up.
  • The shortest sides (NP and JL) match up.
  • PM and KL must match up because they are the only sides left.

Now we will work out what each side in ΔJKL should be multiplied by:

NMJK=3514=2,5MPKL=27,511=2,5NPJL=208=2,5

This means that any side in ΔJKL must be multiplied by 2,5 to make its matching side in ΔNMP.

For example, 14×2,5=35.

We call this number the proportionality constant.

If any one of these division sums had given us a different answer, the sides of the triangles would not have been in proportion. Then, the triangles would not be similar.

TIP: If your calculations tell you that the triangles are not similar in a question where you are asked to prove similarity, you probably made a mistake somewhere. Check your working out!

STEP: Prove similarity
[−1 point ⇒ 0 / 4 points left]

Now that we have matched up the sides, and worked out the proportionality constant, we just need to put the information into a formal similarity proof:

In ΔJKL and ΔNMP:

  1. NM÷JK = 3514 = 2,5
  2. MP÷KL = 27,511 = 2,5
  3. NP÷JL = 208 = 2,5

ΔJKL|||ΔNMP (sides of Δ in prop)


Submit your answer as: andandandandand

Basic deductions from similar triangles

  1. In the diagram below, ΔRQP|||ΔDEF, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • P= (ΔRQP|||ΔDEF)
    • Q= (ΔRQP|||ΔDEF)
    • R= (ΔRQP|||ΔDEF)
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔRQP is similar to ΔDEF, because the sides are in proportion. We can see that the all of the sides in ΔRQP have been multiplied by 2 to get the sides in ΔDEF.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔRQP|||ΔDEF.

    So,

    • R^=D^(ΔRQP|||ΔDEF)
    • Q^=E^(ΔRQP|||ΔDEF)
    • P^=F^(ΔRQP|||ΔDEF)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔCBA|||ΔMNL, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔCBA is similar to ΔMNL, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔCBA|||ΔMNL, we can see that CB matches with MN.

    In the same way, we can see that CA matches with ML (ΔCBA|||ΔMNL).

    Finally, the last sides must also match up. So BA matches with NL (ΔCBA|||ΔMNL).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔCBA's sides in the numerator for the first fraction, then ΔCBA's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    CBMN=BANL=CAML(ΔCBA|||ΔMNL)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. MNCB=NLBA=MLCA(ΔCBA|||ΔMNL)
    2. BANL=CAML=CBMN(ΔCBA|||ΔMNL)

    Submit your answer as:

Basic deductions from similar triangles

  1. In the diagram below, ΔACB|||ΔMNL, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • B= (ΔACB|||ΔMNL)
    • A= (ΔACB|||ΔMNL)
    • C= (ΔACB|||ΔMNL)
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔACB is similar to ΔMNL, because the sides are in proportion. We can see that the all of the sides in ΔACB have been multiplied by 2 to get the sides in ΔMNL.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔACB|||ΔMNL.

    So,

    • A^=M^(ΔACB|||ΔMNL)
    • C^=N^(ΔACB|||ΔMNL)
    • B^=L^(ΔACB|||ΔMNL)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔPRQ|||ΔDEF, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔPRQ is similar to ΔDEF, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔPRQ|||ΔDEF, we can see that PR matches with DE.

    In the same way, we can see that PQ matches with DF (ΔPRQ|||ΔDEF).

    Finally, the last sides must also match up. So RQ matches with EF (ΔPRQ|||ΔDEF).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔDEF's sides in the numerator for the first fraction, then ΔDEF's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    EFRQ=DFPQ=DEPR(ΔPRQ|||ΔDEF)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. RQEF=PQDF=PRDE(ΔPRQ|||ΔDEF)
    2. DFPQ=DEPR=EFRQ(ΔPRQ|||ΔDEF)

    Submit your answer as:

Basic deductions from similar triangles

  1. In the diagram below, ΔMNL|||ΔABC, because their sides are in proportion.

    What can you work out about the angles in the two triangles?

    Answer:
    • N= (ΔMNL|||ΔABC)
    • L= (ΔMNL|||ΔABC)
    • M= (ΔMNL|||ΔABC)
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the sides of two triangles are in proportion, then their matching angles are equal. Can you match up the angles of the two triangles?


    STEP: Match up the angles
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔMNL is similar to ΔABC, because the sides are in proportion. We can see that the all of the sides in ΔMNL have been multiplied by 2 to get the sides in ΔABC.

    If the sides of two triangles are in proportion, then all of their matching angles will be equal. We can identify the matching angles using the similarity statement: ΔMNL|||ΔABC.

    So,

    • M^=A^(ΔMNL|||ΔABC)
    • N^=B^(ΔMNL|||ΔABC)
    • L^=C^(ΔMNL|||ΔABC)
    NOTE: We only know that the angles are equal because the triangles are similar. So, we write this as the reason next to the deduction.

    Submit your answer as: andand
  2. In the diagram below, ΔDFE|||ΔPQR, because their angles are equal.

    What can you deduce about the sides of the two triangles? Choose the correct option.

    Answer:
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    If the angles in two triangles are equal, then their sides will be in proportion. Which of the options shows the correct ratios?


    STEP: Match up the sides
    [−3 points ⇒ 0 / 3 points left]

    We have been told that ΔDFE is similar to ΔPQR, because the matching angles are equal.

    If the matching angles in two triangles are equal, then their sides are in proportion. So, when we divide each side of one triangle by its partner in the other triangle, we always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    Since ΔDFE|||ΔPQR, we can see that DF matches with PQ.

    In the same way, we can see that DE matches with PR (ΔDFE|||ΔPQR).

    Finally, the last sides must also match up. So FE matches with QR (ΔDFE|||ΔPQR).

    TIP: We must always divide the triangle sides in the same order. So, if we put ΔPQR's sides in the numerator for the first fraction, then ΔPQR's sides must always be in the numerator.

    We can also see the matching sides in the diagram:

    So, the correct answer is

    PQDF=PRDE=QRFE(ΔDFE|||ΔPQR)
    NOTE: We only know that the sides are in proportion because the triangles are similar. So, we write this as the reason.

    There are lots of correct ways to write the ratios. These are two more correct examples:

    1. DFPQ=DEPR=FEQR(ΔDFE|||ΔPQR)
    2. PRDE=QRFE=PQDF(ΔDFE|||ΔPQR)

    Submit your answer as:

Identify similar triangles

In the diagram below, B^=36°, and BC^D=BE^A=98°.

Identify one triangle that is similar to ΔEAB.

Answer: ΔEAB|||Δ
one-of
type(string.nocase)
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which other triangle in the diagram has the same angles as ΔEAB?


STEP: Identify the similar triangle
[−2 points ⇒ 0 / 2 points left]

We can see that B^ is common to ΔEAB and ΔCDB. We were also told that BC^D=BE^A.

So, we have two pairs of matching angles in ΔEAB and in ΔCDB.

We can use sum of angles in both triangles to calculate that A^=D^=46°.

So all three pairs of matching angles are equal, and ΔEAB|||ΔCDB.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

Submit your answer as:

Identify similar triangles

In the diagram below, LMMK and MJLK. Also, L^=36° and K^=54°.

Identify one triangle that is similar to ΔLMK.

Answer: ΔLMK|||Δ
one-of
type(string.nocase)
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which other triangle in the diagram has the same angles as ΔLMK?


STEP: Identify the similar triangle
[−2 points ⇒ 0 / 2 points left]

There are three triangles in the diagram: ΔLMK, ΔMJK, and ΔLJM. They are all similar to each other.

Looking at ΔLMK and ΔMJK, we can see that:

  • K^ is a common angle.
  • LM^K=KJ^M=90°, which was given.
  • KM^J=L^=36°, using sum of angles in ΔMJK.

So, ΔLMK|||ΔMJK, because the angles in both triangles are equal.

We could also prove that ΔLMK|||ΔLJM.

Since ΔMJK and ΔLJM are both similar to ΔLMK, they are also similar to each other.

So, either ΔMJK or ΔLJM is a correct answer.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

Submit your answer as:

Identify similar triangles

In the diagram below, R^=45°, and S^=U^=43°.

Identify one triangle that is similar to ΔRSP.

Answer: ΔRSP|||Δ
one-of
type(string.nocase)
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which other triangle in the diagram has the same angles as ΔRSP?


STEP: Identify the similar triangle
[−2 points ⇒ 0 / 2 points left]

We can see that R^ is common to ΔRUT and ΔRSP. We were also told that S^=U^.

So, we have two pairs of matching angles in ΔRUT and in ΔRSP.

We can use sum of angles in both triangles to calculate that RT^U=RP^S=92°.

So all three pairs of matching angles are equal, and ΔRUT|||ΔRSP.

TIP: Whenever we know that two pairs of angles are the same in two triangles, the third pair will automatically also be the same. This is because of the sum of the angles in the two triangles.

Submit your answer as:

Prove similarity with parallel lines

In the diagram below, ADEC.

Prove that ΔADB|||ΔECB.

Answer:

In ΔADB and ΔECB:

  1. BDA= (corresp s; ADEC)
  2. BAD= (corresp s; ADEC)

ΔADB|||Δ .

string
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
[−3 points ⇒ 1 / 4 points left]

There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔADB and ΔECB:

  1. BDA=BCE (corresp s; ADEC)
  2. BAD=BEC (corresp s; ADEC)
  3. B is common

STEP: Complete the proof by labelling the triangles and giving a reason
[−1 point ⇒ 0 / 4 points left]

We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at A is equal to the angle at E. So, Point A matches with Point E.

In the same way:

  • Point D matches with C, and
  • Point B matches with B.

So ΔADB|||ΔECB (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

Prove similarity with parallel lines

In the diagram below, CAEB. Also, CB and AE are straight lines.

Prove that ΔCAD|||ΔBED.

Answer:

In ΔCAD and ΔBED:

  1. CDA=BDE
  2. A=E
  3. C=B

ΔCAD|||Δ .

string
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
[−3 points ⇒ 1 / 4 points left]

There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔCAD and ΔBED:

  1. CDA=BDE (vert opp s equal)
  2. A=E (alt s; CAEB)
  3. C=B (alt s; CAEB)

STEP: Complete the proof by labelling the triangles and giving a reason
[−1 point ⇒ 0 / 4 points left]

We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at C is equal to the angle at B. So, Point C matches with Point B.

In the same way:

  • Point A matches with E, and
  • Point D matches with D.

So ΔCAD|||ΔBED (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

Prove similarity with parallel lines

In the diagram below, SQTR. Also, SR and QT are straight lines.

Prove that ΔSQP|||ΔRTP.

Answer:

In ΔSQP and ΔRTP:

  1. SPQ=RPT
  2. Q=T
  3. S=R

ΔSQP|||Δ .

string
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Two triangles are similar if all of their matching angles are equal.


STEP: Prove that the triangles are equiangular
[−3 points ⇒ 1 / 4 points left]

There are two ways to prove that triangles are similar. We can prove that

  • their matching sides are in proportion (sides of Δ in prop), or
  • their matching angles are equal (equiangular Δs).

We have not been given any information about sides, so we will not be able to prove that the sides of these triangles are in proportion. But, we can use our geometry reasons to prove that their matching angles are equal.

TIP: When you are told that two lines are parallel, you should look out for alternate, corresponding, and co-interior angles.

In ΔSQP and ΔRTP:

  1. SPQ=RPT (vert opp s equal)
  2. Q=T (alt s; SQTR)
  3. S=R (alt s; SQTR)

STEP: Complete the proof by labelling the triangles and giving a reason
[−1 point ⇒ 0 / 4 points left]

We must always match up the triangle vertices on the last line of the congruency proof. This is easier to see when the triangles are drawn separately.

We can see that the angle at S is equal to the angle at R. So, Point S matches with Point R.

In the same way:

  • Point Q matches with T, and
  • Point P matches with P.

So ΔSQP|||ΔRTP (equiangular Δs).

NOTE: You must use the exact reason equiangular Δs. This means that each angle in one triangle is equal to an angle in the other triangle.

Submit your answer as: andandandand

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
[−2 points ⇒ 0 / 2 points left]

There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the sides of all the triangles and so we need to find a diagram that shows a pair of triangles with all three pairs of corresponding sides in proportion.

We look at the side labels. In Diagram A we note the three pairs of corresponding sides are in proportion. In Diagram B we note that the three pairs of corresponding sides are in different proportions.

Therefore Diagram A gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram A, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is A.


Submit your answer as:

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
[−2 points ⇒ 0 / 2 points left]

There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the sides of all the triangles and so we need to find a diagram that shows a pair of triangles with all three pairs of corresponding sides in proportion.

We look at the side labels. In Diagram A we note the three pairs of corresponding sides are in proportion. In Diagram B we note that the three pairs of corresponding sides are in different proportions.

Therefore Diagram A gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram A, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is A.


Submit your answer as:

Similar triangles

Given the following diagrams:

Diagram A
Diagram B

Which diagram shows a pair of similar triangles?

Answer: The similar triangles are in Diagram .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

In maths, the word 'similar' means that two objects have the same shape. Look at the triangles shown and try to determine which pair of triangles are the same shape as each other.


STEP: Compare the triangles and decide which ones are similar
[−2 points ⇒ 0 / 2 points left]

There are two ways we can check for similarity in triangles: using the sides or using the angles.

  1. Angles (AAA): If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar.
  2. Sides (SSS): If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar.

We note that we are given information about the sides of all the triangles and so we need to find a diagram that shows a pair of triangles with all three pairs of corresponding sides in proportion.

We look at the side labels. In Diagram A we note the three pairs of corresponding sides are in proportion. In Diagram B we note that the three pairs of corresponding sides are in different proportions.

Therefore Diagram A gives a pair of triangles that are similar. When two shapes are similar, we use the symbol ||| to show it. For example, in Diagram A, ΔACB|||ΔDFE, which means, 'triangle ACB is similar to triangle DFE.'

The correct choice is A.


Submit your answer as:

Prove simple similarity

Consider the triangles given below:

Select the correct options to prove that ΔYZX|||ΔFED.

Answer:

In ΔYZX and ΔFED:

Step Reason
1. Z=E=81° (given)
2. X=52° (given)
=52°
3. F=47° (given)
=47°
ΔYZX|||ΔFED
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

In ΔZXY:

Y^=180°81°52° (sum of s in Δ)
Y^=47°

In ΔEDF:

D^=180°81°47° (sum of s in Δ)
D^=52°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔYZX and ΔFED:

  1. Z^=E^=81° (given)
  2. X^=52° (given)
    D^=52° (sum of s in Δ)
    X^=D^
  3. F^=47° (given)
    Y^=47° (sum of s in Δ)
    F^=Y^

ΔYZX|||ΔFED (equiangular Δs)


Submit your answer as: andandandandandand

Prove simple similarity

Consider the triangles given below. ZV and YW are straight lines.

Select the correct options to prove that ΔZXW|||ΔYXV.

Answer:

In ΔZXW and ΔYXV:

Step Reason
1. Z=Y=72° (given)
2. YXV=41° (given)
=41°
3. V=67° (sum of s in Δ)
=67°
ΔZXW|||ΔYXV
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

At Point X, two straight lines intersect with each other.

YX^V=ZX^W=41° (vert opp s equal)

In ΔYXV:

V^=180°72°41° (sum of s in Δ)
V^=67°

In ΔZXW:

W^=180°72°41° (sum of s in Δ)
W^=67°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔZXW and ΔYXV:

  1. Z^=Y^=72° (given)
  2. YX^V=41° (given)
    ZX^W=41° (vert opp s equal)
    YX^V=ZX^W
  3. V^=67° (sum of s in Δ)
    W^=67° (sum of s in Δ)
    V^=W^

ΔZXW|||ΔYXV (equiangular Δs)


Submit your answer as: andandandandandand

Prove simple similarity

Consider the triangles given below. SR and TP are straight lines.

Select the correct options to prove that ΔSQP|||ΔTQR.

Answer:

In ΔSQP and ΔTQR:

Step Reason
1. S=T=52° (given)
2. TQR=84° (given)
=84°
3. R=44° (sum of s in Δ)
=44°
ΔSQP|||ΔTQR
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

There are two ways of proving similarity:

  1. Prove that the two triangles are equiangular.
  2. Prove that the two triangles have sides in proportion.

Which would be more relevant here?


STEP: Find missing angles
[−4 points ⇒ 1 / 5 points left]

In these two triangles, we have more information about the angles than about the sides. So, we will prove that the two triangles are equiangular.

In order to do this, first we need to find the missing angles.

At Point Q, two straight lines intersect with each other.

TQ^R=SQ^P=84° (vert opp s equal)

In ΔTQR:

R^=180°52°84° (sum of s in Δ)
R^=44°

In ΔSQP:

P^=180°52°84° (sum of s in Δ)
P^=44°


STEP: Complete the proof
[−1 point ⇒ 0 / 5 points left]

Since we have found three pairs of equal angles, we know that the two triangles are similar. Now we need to write our conclusion as a formal proof:

In ΔSQP and ΔTQR:

  1. S^=T^=52° (given)
  2. TQ^R=84° (given)
    SQ^P=84° (vert opp s equal)
    TQ^R=SQ^P
  3. R^=44° (sum of s in Δ)
    P^=44° (sum of s in Δ)
    R^=P^

ΔSQP|||ΔTQR (equiangular Δs)


Submit your answer as: andandandandandand

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is similar to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other.


STEP: Choose the similar polygon
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other. For two polygons to be similar, they must have:

  • the same number of sides,
  • all of the matching sides in proportion, and
  • all of the matching angles equal in size.
NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Option A is the same shape as the given polygon, even though it is a different size.

So, Option A is similar to the given polygon.


Submit your answer as:

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is similar to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other.


STEP: Choose the similar polygon
[−2 points ⇒ 0 / 2 points left]

Two polygons are similar if they are exactly the same shape, but one might be bigger or smaller than the other. For two polygons to be similar, they must have:

  • the same number of sides,
  • all of the matching sides in proportion, and
  • all of the matching angles equal in size.
NOTE: In normal English, we say two things are similar if they are almost the same. But, in Maths, it is not precise enough to say that similar shapes are approximately the same. In Maths, similarity has a very specific meaning.

Option B is the same shape as the given polygon, even though it is a different size.

So, Option B is similar to the given polygon.


Submit your answer as:

Identifying similar and congruent polygons

Polygon P is drawn below:

Which of the following is congruent to Polygon P?

Answer:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Two polygons are congruent if they are exactly the same size and shape. If we cut out two congruent polygons, one could be placed exactly on top of the other.


STEP: Choose the congruent polygon
[−2 points ⇒ 0 / 2 points left]

Two polygons are congruent if they are exactly the same size and shape. For two polygons to be congruent, they must have:

  • the same number of sides,
  • all of the matching sides equal in length, and
  • all of the matching angles equal in size.
NOTE: If two polygons are the same shape, but different sizes, then they are similar. But they are not congruent. We could not place one polygon exactly on top of the other!

Option D is exactly the same size and shape as the given polygon. This is true even though they are in different orientations.

TIP: Congruent (or similar) polygons can have a different orientation. This just means that they are facing different directions. If we cut them out and turn them we could still place one directly on top of the other.

Submit your answer as:

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of m.

Answer: m= units
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
[−2 points ⇒ 0 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 12 to give the matching sides in the second polygon. For example:

  • 10×12=5
  • 12×12=6

The side labelled m matches with the side equal to 8 in the first polygon. So:

8×12=mm=4 units
NOTE: We can see that the sides in the first polygon have all been divided by 2 to get the sides in the second polygon. Dividing by 2 is the same as multiplying by 12.

Submit your answer as:

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of a.

Answer: a= units
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
[−2 points ⇒ 0 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 12 to give the matching sides in the second polygon. For example:

  • 18×12=9
  • 12×12=6

The side labelled a matches with the side equal to 16 in the first polygon. So:

16×12=aa=8 units
NOTE: We can see that the sides in the first polygon have all been divided by 2 to get the sides in the second polygon. Dividing by 2 is the same as multiplying by 12.

Submit your answer as:

Calculate a side in similar polygons

The two polygons drawn below are similar.

Determine the value of n.

Answer: n= units
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon.


STEP: Determine the scale factor
[−2 points ⇒ 0 / 2 points left]

If two polygons are similar, then their sides are in proportion. This means that every side in the first polygon has been multiplied by the same number to get the matching sides in the second polygon. This number is called the scale factor.

The sides in the first polygon have been multiplied by 2 to give the matching sides in the second polygon. For example:

  • 10×2=20
  • 4×2=8

The side labelled n matches with the side equal to 5 in the first polygon. So:

5×2=nn=10 units

Submit your answer as:

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Erioluwa needs to prove that ΔYXZ|||ΔABC. She has already started, and her incomplete proof is written below:

    In ΔYXZ and ΔABC:

    1. Y^=A^ (given)
      ...
    Answer:

    How should Erioluwa complete her proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option B

    This "proof" makes mistakes on Step 2 and Step 3. We were not given that X^=B^; we must prove this using sum of s in Δ. Also, on Step 3, we know that C^=51° because this was given , not because of sum of s in Δ.

    Option C

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option D

    We cannot prove that the triangles are congruent, even though we can prove a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. The information we have been given actually proves that the triangles are slightly different sizes.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔYXZ and ΔABC:

    1. Y^=A^ (given)
    2. Z^=51° (sum of s in Δ)
      Z^=C^
    3. Also, B^=40° (sum of s in Δ)
      X^=B^

    ΔYXZ|||ΔABC (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of AB, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: AB=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for AB. So, start by writing AB in the numerator:

    AB=

    Since ΔYXZ|||ΔABC, we can see that AB matches with YX.

    So we have:

    ABYX=

    We were also given information about CA and YZ. Since ΔYXZ|||ΔABC, we can see that CA matches with YZ.

    So,

    ABYX=CAYZ(ΔYXZ|||ΔABC)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    AB12=1210AB12×12=1210×12AB=725=14,4

    Submit your answer as: and

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Eniola needs to prove that ΔMNL|||ΔBCA. She has already started, and her incomplete proof is written below:

    In ΔMNL and ΔBCA:

    1. M^=B^ (given)
      ...
    Answer:

    How should Eniola complete her proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    This "proof" makes mistakes on Step 2 and Step 3. We were not given that N^=C^; we must prove this using sum of s in Δ. Also, on Step 3, we know that A^=84° because this was given , not because of sum of s in Δ.

    Option B

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option C

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option D

    We cannot prove that the triangles are congruent, even though we can prove a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. The information we have been given actually proves that the triangles are slightly different sizes.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔMNL and ΔBCA:

    1. M^=B^ (given)
    2. L^=84° (sum of s in Δ)
      L^=A^
    3. Also, C^=41° (sum of s in Δ)
      N^=C^

    ΔMNL|||ΔBCA (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of BC, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: BC=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for BC. So, start by writing BC in the numerator:

    BC=

    Since ΔMNL|||ΔBCA, we can see that BC matches with MN.

    So we have:

    BCMN=

    We were also given information about AB and ML. Since ΔMNL|||ΔBCA, we can see that AB matches with ML.

    So,

    BCMN=ABML(ΔMNL|||ΔBCA)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    BC15=1510BC15×15=1510×15BC=452=22,5

    Submit your answer as: and

Similarity or congruency?

Consider the diagram below and answer the questions that follow:

  1. Nnenne needs to prove that ΔSTU|||ΔFDE. She has already started, and her incomplete proof is written below:

    In ΔSTU and ΔFDE:

    1. S^=F^ (given)
      ...
    Answer:

    How should Nnenne complete her proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Two triangles are similar if they are the same shape, but they may be a different size. Their angles will be equal and their sides will be in proportion.

    Two triangles are congruent if they are exactly the same size and shape. Their matching sides and angles will all be equal.


    STEP: Choose the correct proof
    [−4 points ⇒ 0 / 4 points left]

    Option A

    We cannot prove that the triangles are congruent, even though we can prove a pair of equal sides and two pairs of equal angles. This is because the angles are not in the same positions relative to the given side. The information we have been given actually proves that the triangles are slightly different sizes.

    Option B

    To prove that two triangles are similar, you must either prove that all of the sides are in proportion, or that all the matching angles are equal . This proof tries to do a mixture of these two, which is not allowed.

    Option C

    This option uses sum of s in Δ correctly to prove that there are three pairs of matching angles in the triangles.

    Option D

    This "proof" makes mistakes on Step 2 and Step 3. We were not given that T^=D^; we must prove this using sum of s in Δ. Also, on Step 3, we know that E^=92° because this was given , not because of sum of s in Δ.

    TIP: The question told us to prove that the triangles are similar (using the symbol |||). You should always read the question statement very carefully.

    The similarity is easier to see when the triangles are in the same orientation:

    The full proof is shown below:

    In ΔSTU and ΔFDE:

    1. S^=F^ (given)
    2. U^=92° (sum of s in Δ)
      U^=E^
    3. Also, D^=45° (sum of s in Δ)
      T^=D^

    ΔSTU|||ΔFDE (equiangular Δs)


    Submit your answer as:
  2. Hence determine the length of FD, giving a reason for your answer.

    INSTRUCTION: Do not round off your answer.
    Answer: FD=
    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Use the similarity statement to choose the correct pair of ratios.


    STEP: Use the similarity statement to choose the correct pair of ratios
    [−2 points ⇒ 1 / 3 points left]

    If two triangles are similar, their sides are in proportion. This means that when you divide the matching sides by each other, you always get the same number.

    TIP: Use the similarity statement to choose the correct pair of ratios.

    We are looking for FD. So, start by writing FD in the numerator:

    FD=

    Since ΔSTU|||ΔFDE, we can see that FD matches with ST.

    So we have:

    FDST=

    We were also given information about EF and SU. Since ΔSTU|||ΔFDE, we can see that EF matches with SU.

    So,

    FDST=EFSU(ΔSTU|||ΔFDE)
    NOTE: We only know that the fractions are equal because the triangles are similar. So we write this as the reason. Remember, sides of Δ in prop is used to prove that triangles are similar, not to determine side lengths.

    STEP: Solve the equation using inverse operations
    [−1 point ⇒ 0 / 3 points left]
    FD14=1410FD14×14=1410×14FD=985=19,6

    Submit your answer as: and

2. Scale factors

Consequences of similarity with calculations

In the diagram below, ΔABC|||ΔDEF.

  1. Determine the size of D^, in terms of x.

    Answer:

    A^= °
    D^= °

    expression
    expression
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate D^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔABC
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate D^ in one step, because we do not have enough information in ΔDEF.

    But, we do have enough information in ΔABC to calculate A^. Knowing the size of A^ will help us to calculate D^.

    In ΔABC:

    A^+(x+25°)+90°=180°(sum of s in Δ)
    A^=180°90°x25°A^=65°x

    STEP: Compare matching angles in ΔDEF
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔDEF, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate D^.

    We were told that ΔABC|||ΔDEF. So, A^ and D^ are matching angles, because their letters come first. Similarly, B^ and E^ are matching angles. Finally, C^ and F^ are matching angles.

    A^ matches D^. So,

    D^=65°x(ΔABC|||ΔDEF)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of AB.

    Answer:

    DE= units
    AB= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate AB in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔDEF
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate AB in one step, because we do not have enough information in ΔABC.

    But, we do have enough information in ΔDEF to calculate DE. Then we can use the fact that the triangles are similar to calculate AB.

    In ΔDEF:

    DE2=62+2,52(Pythagoras)DE2=42,25DE=6,5
    NOTE: We can only use the theorem of Pythagoras because ΔDEF is right-angled!

    STEP: Compare matching sides in ΔABC
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔDEF, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate AB.

    We were told that ΔABC|||ΔDEF. So, AB and DE are matching sides, because their letters came first. In the same way, BC and EF are matching sides because their letters came last. And AC and DF are matching sides, because they are the only sides left.

    We know the lengths of EF and BC, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔDEF by to get the sides of ΔABC. So, we divide the known side of ΔABC by the known side of ΔDEF.

    k=BCFE=126=2

    The sides are in proportion, so when we multiply any side in ΔDEF by 2, we will get the matching side in ΔABC.

    AB=DE×2(ΔABC|||ΔDEF)AB=6,5×2=13

    Submit your answer as: andandand

Exercises

Consequences of similarity with calculations

In the diagram below, ΔABC|||ΔDEF.

  1. Determine the size of D^, in terms of x.

    Answer:

    A^= °
    D^= °

    expression
    expression
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate D^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔABC
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate D^ in one step, because we do not have enough information in ΔDEF.

    But, we do have enough information in ΔABC to calculate A^. Knowing the size of A^ will help us to calculate D^.

    In ΔABC:

    A^+(x+30°)+90°=180°(sum of s in Δ)
    A^=180°90°x30°A^=60°x

    STEP: Compare matching angles in ΔDEF
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔDEF, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate D^.

    We were told that ΔABC|||ΔDEF. So, A^ and D^ are matching angles, because their letters come first. Similarly, B^ and E^ are matching angles. Finally, C^ and F^ are matching angles.

    A^ matches D^. So,

    D^=60°x(ΔABC|||ΔDEF)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of AB.

    Answer:

    DE= units
    AB= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate AB in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔDEF
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate AB in one step, because we do not have enough information in ΔABC.

    But, we do have enough information in ΔDEF to calculate DE. Then we can use the fact that the triangles are similar to calculate AB.

    In ΔDEF:

    DE2=82+62(Pythagoras)DE2=100DE=10
    NOTE: We can only use the theorem of Pythagoras because ΔDEF is right-angled!

    STEP: Compare matching sides in ΔABC
    [−2 points ⇒ 0 / 4 points left]

    Since ΔABC|||ΔDEF, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate AB.

    We were told that ΔABC|||ΔDEF. So, AB and DE are matching sides, because their letters came first. In the same way, BC and EF are matching sides because their letters came last. And AC and DF are matching sides, because they are the only sides left.

    We know the lengths of EF and BC, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔDEF by to get the sides of ΔABC. So, we divide the known side of ΔABC by the known side of ΔDEF.

    k=BCFE=168=2

    The sides are in proportion, so when we multiply any side in ΔDEF by 2, we will get the matching side in ΔABC.

    AB=DE×2(ΔABC|||ΔDEF)AB=10×2=20

    Submit your answer as: andandand

Consequences of similarity with calculations

In the diagram below, ΔQRS|||ΔYXZ.

  1. Determine the size of Y^, in terms of x.

    Answer:

    Q^= °
    Y^= °

    expression
    expression
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate Y^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔQRS
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate Y^ in one step, because we do not have enough information in ΔYXZ.

    But, we do have enough information in ΔQRS to calculate Q^. Knowing the size of Q^ will help us to calculate Y^.

    In ΔQRS:

    Q^+(x+20°)+90°=180°(sum of s in Δ)
    Q^=180°90°x20°Q^=70°x

    STEP: Compare matching angles in ΔYXZ
    [−2 points ⇒ 0 / 4 points left]

    Since ΔQRS|||ΔYXZ, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate Y^.

    We were told that ΔQRS|||ΔYXZ. So, Q^ and Y^ are matching angles, because their letters come first. Similarly, R^ and X^ are matching angles. Finally, S^ and Z^ are matching angles.

    Q^ matches Y^. So,

    Y^=70°x(ΔQRS|||ΔYXZ)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of QR.

    Answer:

    YX= units
    QR= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate QR in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔYXZ
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate QR in one step, because we do not have enough information in ΔQRS.

    But, we do have enough information in ΔYXZ to calculate YX. Then we can use the fact that the triangles are similar to calculate QR.

    In ΔYXZ:

    YX2=62+4,52(Pythagoras)YX2=56,25YX=7,5
    NOTE: We can only use the theorem of Pythagoras because ΔYXZ is right-angled!

    STEP: Compare matching sides in ΔQRS
    [−2 points ⇒ 0 / 4 points left]

    Since ΔQRS|||ΔYXZ, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate QR.

    We were told that ΔQRS|||ΔYXZ. So, QR and YX are matching sides, because their letters came first. In the same way, RS and XZ are matching sides because their letters came last. And QS and YZ are matching sides, because they are the only sides left.

    We know the lengths of XZ and RS, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔYXZ by to get the sides of ΔQRS. So, we divide the known side of ΔQRS by the known side of ΔYXZ.

    k=RSZX=126=2

    The sides are in proportion, so when we multiply any side in ΔYXZ by 2, we will get the matching side in ΔQRS.

    QR=YX×2(ΔQRS|||ΔYXZ)QR=7,5×2=15

    Submit your answer as: andandand

Consequences of similarity with calculations

In the diagram below, ΔJKL|||ΔPMN.

  1. Determine the size of P^, in terms of x.

    Answer:

    J^= °
    P^= °

    expression
    expression
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate P^ in one step. Are there any other angles that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing angle in ΔJKL
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate P^ in one step, because we do not have enough information in ΔPMN.

    But, we do have enough information in ΔJKL to calculate J^. Knowing the size of J^ will help us to calculate P^.

    In ΔJKL:

    J^+(x+30°)+90°=180°(sum of s in Δ)
    J^=180°90°x30°J^=60°x

    STEP: Compare matching angles in ΔPMN
    [−2 points ⇒ 0 / 4 points left]

    Since ΔJKL|||ΔPMN, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for an angle. So, we will use the fact that matching angles are equal between similar triangles to calculate P^.

    We were told that ΔJKL|||ΔPMN. So, J^ and P^ are matching angles, because their letters come first. Similarly, K^ and M^ are matching angles. Finally, L^ and N^ are matching angles.

    J^ matches P^. So,

    P^=60°x(ΔJKL|||ΔPMN)

    Submit your answer as: andandand
  2. Now, use the same diagram to determine the length of JL.

    Answer:

    PN= units
    JL= units

    numeric
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    In similar triangles,

    1. angles are equal, and
    2. sides are in proportion.

    You do not have enough information to calculate JL in one step. Are there any other sides that you can calculate?

    Then, you can use the fact that the triangles are similar.


    STEP: Calculate the missing side in ΔPMN
    [−2 points ⇒ 2 / 4 points left]

    We cannot calculate JL in one step, because we do not have enough information in ΔJKL.

    But, we do have enough information in ΔPMN to calculate PN. Then we can use the fact that the triangles are similar to calculate JL.

    In ΔPMN:

    302=242+PN2(Pythagoras)324=PN218=PN
    NOTE: We can only use the theorem of Pythagoras because ΔPMN is right-angled!

    STEP: Compare matching sides in ΔJKL
    [−2 points ⇒ 0 / 4 points left]

    Since ΔJKL|||ΔPMN, we know that the two triangles are equiangular and have sides in proportion.

    We are looking for a side. So, we will use the fact that sides in similar triangles are in proportion to calculate JL.

    We were told that ΔJKL|||ΔPMN. So, JK and PM are matching sides, because their letters came first. In the same way, KL and MN are matching sides because their letters came last. And JL and PN are matching sides, because they are the only sides left.

    We know the lengths of MN and KL, so we will use them to calculate the value of the proportionality constant.

    We want to know what to multiply the sides of ΔPMN by to get the sides of ΔJKL. So, we divide the known side of ΔJKL by the known side of ΔPMN.

    k=KLNM=1224=0,5

    The sides are in proportion, so when we multiply any side in ΔPMN by 0,5, we will get the matching side in ΔJKL.

    JL=PN×0,5(ΔJKL|||ΔPMN)JL=18×0,5=9

    Submit your answer as: andandand

3. Practical applications

Applications of similarity

Akeju wants to find out the height of a telephone pole. Akeju is 1,54 m tall. The length of Akeju's shadow changes throughout the day.

At 3 pm, Akeju stands in the shadow of the telephone pole. The shadows of the telephone pole and Akeju form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 3 pm, Akeju's shadow is 2,67 m long and the telephone pole's shadow is 5,87 m long. Complete the proof to help Akeju work out the height of the telephone pole.

    Answer:

    In ΔJKL and ΔPNL:

    1. is common.
    2. J=LPN= ° (because the telephone pole and Akeju are both perpendicular to the ground)
    3. K=LNP= ° (sum of s in Δ)

    ΔJKL|||ΔPNL

    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like telephone poles and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at KJ^L and NP^L are 90°, we can use sum of s in Δ to work out expressions for the angles at JK^L and PN^L.

    In ΔJKL:

    J^+K^+L^=180°(sum of s in Δ)90°+K^+θ=180°90°+K^+θ90°θ=180°90°θK^=90°θ

    Similarly, in ΔPNL:

    P^+N^+L^=180°(sum of s in Δ)90°+N^+θ=180°N^=90°θ

    So, even though we don't know what θ is, we can say for certain that K^=N^, because they are both equal to 90°θ.

    For example, if θ=30°, K^ and N^ would both be 60°. If θ=50°, K^ and N^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔJKL and ΔPNL:

    1. L is common.
    2. J=P=90° (because the telephone pole and Akeju are both perpendicular to the ground)
    3. K=N=90°θ (sum of s in Δ)

    ΔJKL|||ΔPNL (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 3 pm, Akeju's shadow is 2,67 m long and the telephone pole's shadow is 5,87 m long. We have also proved that ΔJKL|||ΔPNL.

    Hence, work out the height of the telephone pole.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the telephone pole = m (ΔJKL|||ΔPNL)

    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔJKL|||ΔPNL, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the telephone pole (JK) is what we want to find out, we will put it in the numerator. It matches with Akeju's height (PN) in the smaller triangle.

    JK1,54=

    In the same way, the telephone pole's shadow (JL) matches with Akeju's shadow (PL) in the smaller triangle.

    JK1,54=5,872,67

    Hence,

    JK1,54×1,54=5,872,67×1,54JK=3,38569... m3,39 m

    Submit your answer as:

Exercises

Applications of similarity

Nkiru wants to find out the height of a telephone pole. Nkiru is 1,66 m tall. The length of Nkiru's shadow changes throughout the day.

At 3 pm, Nkiru stands in the shadow of the telephone pole. The shadows of the telephone pole and Nkiru form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 3 pm, Nkiru's shadow is 2,88 m long and the telephone pole's shadow is 6,62 m long. Complete the proof to help Nkiru work out the height of the telephone pole.

    Answer:

    In ΔPQR and ΔUTR:

    1. is common.
    2. P=RUT= ° (because the telephone pole and Nkiru are both perpendicular to the ground)
    3. Q=RTU= ° (sum of s in Δ)

    ΔPQR|||ΔUTR

    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like telephone poles and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at QP^R and TU^R are 90°, we can use sum of s in Δ to work out expressions for the angles at PQ^R and UT^R.

    In ΔPQR:

    P^+Q^+R^=180°(sum of s in Δ)90°+Q^+θ=180°90°+Q^+θ90°θ=180°90°θQ^=90°θ

    Similarly, in ΔUTR:

    U^+T^+R^=180°(sum of s in Δ)90°+T^+θ=180°T^=90°θ

    So, even though we don't know what θ is, we can say for certain that Q^=T^, because they are both equal to 90°θ.

    For example, if θ=30°, Q^ and T^ would both be 60°. If θ=50°, Q^ and T^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔPQR and ΔUTR:

    1. R is common.
    2. P=U=90° (because the telephone pole and Nkiru are both perpendicular to the ground)
    3. Q=T=90°θ (sum of s in Δ)

    ΔPQR|||ΔUTR (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 3 pm, Nkiru's shadow is 2,88 m long and the telephone pole's shadow is 6,62 m long. We have also proved that ΔPQR|||ΔUTR.

    Hence, work out the height of the telephone pole.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the telephone pole = m (ΔPQR|||ΔUTR)

    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔPQR|||ΔUTR, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the telephone pole (PQ) is what we want to find out, we will put it in the numerator. It matches with Nkiru's height (UT) in the smaller triangle.

    PQ1,66=

    In the same way, the telephone pole's shadow (PR) matches with Nkiru's shadow (UR) in the smaller triangle.

    PQ1,66=6,622,88

    Hence,

    PQ1,66×1,66=6,622,88×1,66PQ=3,81569... m3,82 m

    Submit your answer as:

Applications of similarity

Faruq wants to find out the height of a flagpole. Faruq is 1,52 m tall. The length of Faruq's shadow changes throughout the day.

At 3 pm, Faruq stands in the shadow of the flagpole. The shadows of the flagpole and Faruq form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 3 pm, Faruq's shadow is 2,63 m long and the flagpole's shadow is 6,31 m long. Complete the proof to help Faruq work out the height of the flagpole.

    Answer:

    In ΔUVW and ΔZYW:

    1. is common.
    2. U=WZY= ° (because the flagpole and Faruq are both perpendicular to the ground)
    3. V=WYZ= ° (sum of s in Δ)

    ΔUVW|||ΔZYW

    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like flagpoles and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at VU^W and YZ^W are 90°, we can use sum of s in Δ to work out expressions for the angles at UV^W and ZY^W.

    In ΔUVW:

    U^+V^+W^=180°(sum of s in Δ)90°+V^+θ=180°90°+V^+θ90°θ=180°90°θV^=90°θ

    Similarly, in ΔZYW:

    Z^+Y^+W^=180°(sum of s in Δ)90°+Y^+θ=180°Y^=90°θ

    So, even though we don't know what θ is, we can say for certain that V^=Y^, because they are both equal to 90°θ.

    For example, if θ=30°, V^ and Y^ would both be 60°. If θ=50°, V^ and Y^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔUVW and ΔZYW:

    1. W is common.
    2. U=Z=90° (because the flagpole and Faruq are both perpendicular to the ground)
    3. V=Y=90°θ (sum of s in Δ)

    ΔUVW|||ΔZYW (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 3 pm, Faruq's shadow is 2,63 m long and the flagpole's shadow is 6,31 m long. We have also proved that ΔUVW|||ΔZYW.

    Hence, work out the height of the flagpole.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the flagpole = m (ΔUVW|||ΔZYW)

    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔUVW|||ΔZYW, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the flagpole (UV) is what we want to find out, we will put it in the numerator. It matches with Faruq's height (ZY) in the smaller triangle.

    UV1,52=

    In the same way, the flagpole's shadow (UW) matches with Faruq's shadow (ZW) in the smaller triangle.

    UV1,52=6,312,63

    Hence,

    UV1,52×1,52=6,312,63×1,52UV=3,64684... m3,65 m

    Submit your answer as:

Applications of similarity

Umar wants to find out the height of a tree. Umar is 1,56 m tall. The length of Umar's shadow changes throughout the day.

At 11 am, Umar stands in the shadow of the tree. The shadows of the tree and Umar form triangles. Both triangles contain the same angle, as shown in the diagram below.

  1. At 11 am, Umar's shadow is 2,7 m long and the tree's shadow is 6,21 m long. Complete the proof to help Umar work out the height of the tree.

    Answer:

    In ΔJKL and ΔPNL:

    1. is common.
    2. J=LPN= ° (because the tree and Umar are both perpendicular to the ground)
    3. K=LNP= ° (sum of s in Δ)

    ΔJKL|||ΔPNL

    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    It may help you to redraw the diagram with the two triangles separate from each other. Which pairs of angles are equal between the two triangles, and why?


    STEP: Prove similarity
    [−4 points ⇒ 0 / 4 points left]
    TIP: In real-life questions like this one, we can always assume that vertical objects like trees and human beings are perpendicular to the ground. We also assume that the ground is perfectly horizontal!

    Once we know that the angles at KJ^L and NP^L are 90°, we can use sum of s in Δ to work out expressions for the angles at JK^L and PN^L.

    In ΔJKL:

    J^+K^+L^=180°(sum of s in Δ)90°+K^+θ=180°90°+K^+θ90°θ=180°90°θK^=90°θ

    Similarly, in ΔPNL:

    P^+N^+L^=180°(sum of s in Δ)90°+N^+θ=180°N^=90°θ

    So, even though we don't know what θ is, we can say for certain that K^=N^, because they are both equal to 90°θ.

    For example, if θ=30°, K^ and N^ would both be 60°. If θ=50°, K^ and N^ would both be 40°. Since they both have the same algebraic expression, they will both have the same value no matter what value we substitute for θ.

    The completed proof is:

    In ΔJKL and ΔPNL:

    1. L is common.
    2. J=P=90° (because the tree and Umar are both perpendicular to the ground)
    3. K=N=90°θ (sum of s in Δ)

    ΔJKL|||ΔPNL (equiangular Δs)


    Submit your answer as: andandand
  2. We have been told that at 11 am, Umar's shadow is 2,7 m long and the tree's shadow is 6,21 m long. We have also proved that ΔJKL|||ΔPNL.

    Hence, work out the height of the tree.

    INSTRUCTION: Round your answer to two decimal places.
    Answer:

    Height of the tree = m (ΔJKL|||ΔPNL)

    one-of
    type(numeric.abserror(0.005))
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    Since the two triangles are similar, their sides are in proportion.


    STEP: Use sides in proportion to create and solve an equation
    [−3 points ⇒ 0 / 3 points left]

    Since we have proved that ΔJKL|||ΔPNL, their sides must be in proportion. We can use the ratio of sides to set up an equation.

    Since the height of the tree (JK) is what we want to find out, we will put it in the numerator. It matches with Umar's height (PN) in the smaller triangle.

    JK1,56=

    In the same way, the tree's shadow (JL) matches with Umar's shadow (PL) in the smaller triangle.

    JK1,56=6,212,7

    Hence,

    JK1,56×1,56=6,212,7×1,56JK=3,588 m3,59 m

    Submit your answer as: